PHL111 Valid Reasoning, past assignments



Past Assignments
September 2.
Read sections 1.1 and 1.2, pages 2-13 of K&M&M's Logic.


September 9
Reading and homework.

Read sections 1.3 and 1.4, pages 13-35.

Homework 1 due: translate the following sentences from English into our logical language as we have developed it so far. Provide one key for the entire assignment that shows to which (hopefully atomic) English sentences your FOL sentence letters correspond; put the one key at the top. These are the sentences:
  1. Smith is a dog.
  2. Smith is not a dog.
  3. Smith is a mammal.
  4. Smith is a reptile.
  5. If Smith is a dog, then Smith is a mammal.
  6. Smith is a dog, if Smith is a mammal.
  7. Smith is not a dog, on the condition that Smith is not a mammal.
  8. Provided that Smith is a mammal, then Smith is not a reptile.
  9. Smith is a reptile only if Smith is a chordate.
  10. Smith is not a chordate.
Hey, be careful. Look in your book, where it explains that there are some alternatives in English equivalent to "if... then...." For example, all of the following are translated 'P-->Q' in our logic: "if P, Q", "Q if P", "provided that P, Q", "Q provided that P", "Q on the condition that P", "on the condition that P, Q" and "P only if Q". That last one always surprises people -- we'll talk about it some more. See page 11 of your text.

The mean was a 7.8 and the standard deviation was 1.6.

(For those who did the wrong homework -- "tentative" means tentative! -- a correct translation would be:
  1. P
  2. ~P
  3. Q
  4. (P --> Q)
  5. (~Q --> ~P)
  6. R
  7. (Q --> ~R)
  8. (R --> S)
  9. ~S
  10. (Q --> P)
)
21 September
Finish reading chapter 1 of Kalish, Montague, and Marr. Complete a direct derivation (also called a direct proof) for each of the following arguments, showing that it is valid.
  1. Premises: (~R --> S), ~R. Show: S.
  2. Premises: (P --> Q), (Q --> R), ~~P. Show: ~~R.
  3. Premises: (T --> P), (Q --> S), (S --> T), ~P. Show: ~Q.
  4. Premises: R, S, (R --> (S --> T)). Show: T.
  5. Premises: (~(P --> S) --> R), ((P --> S) --> Q), ~Q. Show: R.
For this assignment, let's be very precise about the rules, and not make leaps based upon your understanding.

Mean was 7.7, the standard deviation was 3.1.


September 30
Please complete the following problems. Translate the sentences 1-5 into our logical language. Provide a single key for all five problems. Create a truth table to show the argument given as #6 is valid; circle all and only the rows of the truth table that show the argument is valid. Create a truth table to show the argument given as #7 is invalid; circle all and only the rows of the truth table that show the argument is invalid. Provide proofs of the arguments for #8, 9, and 10.
  1. Both Smith and Jones got bit by the dog.
  2. Both Smith and Jones will need to get the rabbies shot only if they both get bit by the dog.
  3. Smith got bit by the dog and Jones got bit by the cat.
  4. Not both Smith and Jones were bitten by the cat.
  5. If Smith was bitten by the cat and dog both, then Jones was too.
  6. Premises: (P --> Q), ~Q. Conclusion (~P ^ ~Q)
  7. Premises: (P --> Q), Q. Conclusion: P.
  8. Premises: (P ^ Q), (P --> R), (Q --> S). Conclusion: (R ^ S)
  9. Premises: S, T, ((S ^ T) --> V). Conclusion: V.
  10. Premises: R, S, (P --> ~(R ^ S)). Conclusion: ~P.
Yikes. I screwed up and mis-stated problem #6. Sorry. But I treated the homework as out of 9 points (even if I wrote on your homework "x/10") and so it didn't count.
9 October
Complete the following problems and hand them in at the beginning of class. Problems 1-5 require you to translate sentences from English into our language. Use a single translation key. Problems 6-10 require you to do proofs using our inference rules. Problems 6 and 7 can be done with a direct proof. Problems 8-10 require a conditional proof.
  1. Whales and dolphins are mammals if and only if they both nurse their young.
  2. Whales or sharks, but not both, are mammals.
  3. Whales are mammals just in case dolphins are.
  4. Dolphins nurse their young but they also swin in the sea.
  5. Whales nurse their young and sharks don't.
  6. Premises: ~(T --> R), ((P --> V) --> (T --> R)). Show: ~(P --> V).
  7. Premises: (P <--> R), (R ^ ~S), (S v T). Show: ((Q v P) ^ T).
  8. Premises: (Q --> ~R), (~P --> R). Show: (~P --> ~Q).
  9. Premises: (S --> R), (P --> S). Show: (~R --> ~P).
  10. Show: [(P ^ (P --> Q)) --> Q]
Extras
Some of you have asked me for extra problems. Here are a few. In a few days I'll post answers to these (I won't be collecting them).
  1. Premises: (~~P --> S), P. Show: ~~S.
  2. Premises: (~P --> V), ~V, (P --> Q). Show: Q.
  3. Premises: (~(R --> S) --> V), ((R --> S) --> Q), ~Q. Show: V.
  4. Premises: (P --> R), (R --> Q). Show: (P --> Q).
  5. Premises: (S --> R), (P --> S). Show: (~R --> ~P).
  6. #18 in the book, page 27. Premises: ~Q. Show: ((P --> Q) --> ~P).
  7. #21 in the book, page 27. Premises: (Q --> ~R), (~P --> R). Show: (~P --> ~Q).
  8. Premises: (Q --> ~R), (~S --> P). Show: ((P --> Q) --> (R --> S).
  9. #19 in the book, page 27. Premises: (P --> (Q --> R)), (P --> (R --> S)). Show: (P --> (Q --> S)).
  10. Show: R. Premises: ((T v R) <--> S), (S ^ ~Q), (T <--> Q).
  11. Make the truth table for the following sentence: (~(P v Q) <--> (~P ^ ~Q)).
  12. Make the truth table for the following sentence: (~(P ^ Q) <--> (~P v ~Q)).)).
Here are some solutions.

October 14
Exam 1. Definition of validity, theorem, tautology. Translations. Truth tables. Semantics of all connectives. Direct proofs with conditionals, negations, conjunctions. Conditional proofs.

The mean ("average") was 77. The standard deviation was 20. The high was 100; seven students got 100/100. The low was 18.
October 23

NOTE that I pushed the due date ahead 2 days since I had only limited office hours on Monday.
Due at the beginning of class, complete the following proofs. Number 5 requires an indirect proof. Number 6 requires a a condition proof that contains an indirect proof. Number 7 requires you do a direct proof that contains two conditional proofs, and then use CB.
  1. Show: (S --> T). Premises: (~S v P), (R ^ Q), ((P ^ R) --> T).
  2. Show: (P ^ Q). Premises: (~P --> T), (R ^ ~T), (~R v Q).
  3. Show: (R v T). Premises: (S --> R), (T ^ S).
  4. Prove theorem T26, page 108: ((P --> Q) ^ (Q --> R)) --> (P --> R)
  5. Show: ~R. Premises: (Q v ~R), (Q --> S), (R <--> ~S)
  6. Prove theorem T67, page 110: ((~P ^ ~Q) --> ~(P v Q))
  7. Show: (P <--> Q). Premises: (P --> S), (~S v Q), (~P --> ~Q).

November 2
Please complete the following problems. They stretch you a bit by asking you to apply propositional logic. The handout we had on scientific method is available also here. Your answers to 1 and 2 will be largely just written out in English -- that is, I'm not expecting you to symbolize these somehow.
  1. Consider the following account of Semmelweis's discovery that unclean hands can cause disease, as recounted by the philosopher Carl Hempel:
    Ignaz Semmelweis... did this work during the years from 1844 to 1848 at the Vienna General Hospital. As a member of the medical staff of the First Maternity Division in the hospital, Semmelweis was distressed to find that a large proportion of the woemn who were delivered of their babies in that division contracted a serious and often fatal illness.... In the adjacent Second Maternity Division of the same hospital ... the death toll was much lower.... Semmelweis came to the conclusion that his patients had died [because] he, his colleagues, and his medical students were carriers of infectious material, for he and his associates used to come to the wards directly from performing dissections in the autopsy room, and examine the women in labor after only superficially washing their hands, which often retained a characteristic foul odor.
    Describe a general hypothesis H that Semmelweis should form (e.g., "Women patients are dying at a much higher rate in the First Maternity Division because..."), and also a particular testable prediction P that he could conclude is implied by that hypothesis (you should be able to make a conditional from your hypothesis H to prediction P). Each should be a single sentence (that is, one sentence for the hypothesis and one sentence for the prediction). Note that Semmelweis wanted to be ethical, so he would not have proposed a test like dipping babies into autopsies. Rather, presumably his hope is that he could identify some way to reduce deaths, if his hypothesis is true. Make a truth table for your conditional, and explain what Semmelweis would discover if his prediction would prove true. Use the truth table to illustrate this. Remember that we assume your conditional is true so ignore the kind of situation in which it is false.
  2. The Duhem hypothesis is that we actually test a number of hypotheses at once, not just the general one we aim to test. Can you suggest some of the other hypotheses that Semmelweis must assume for your conditional above (the conditional from hypothesis to prediction) to be reasonable? That is, for example, assume that his prediction proves false in one case -- what might have gone wrong such that his main hypothesis could be true even though the prediction turned out false (this would be one of the additional predictions). List at least three.
  3. Make up an argument of at least three premises that is valid, where none of the premises is the conclusion alone, and where all the premises are needed to prove the conclusion. Give the argument both in English, and also in our propositional logic. Provide a key so that I can see how your translation works. Prove the argument it in the propositional logic using a syntactic proof (not a truth table).
  4. Make up an argument of at least five premises that is valid, where none of the premises is the conclusion alone, and where all the premises are needed to prove the conclusion. Try to see if you can do it with a conclusion that is a conditional, so that your proof will be a conditional derivation. Give the argument both in English, and also in our propositional logic. Provide a translation key. Prove the argument in the propositional logic using a syntactic proof (not a truth table).

Alternate/followup homework to November 2
  1. You may do a problem like the one above from any science you like. But if you are stumped, here's an example you can use: When Galileo made his telescope, it was official doctrine and widespread belief that the Earth was the center of the universe and all things revolved around the Earth. Galileo saw to one side of Jupiter, with his telescope, a bright light or seeming star. Each time he looked later, it had moved slightly closer to Jupiter. (We know now that what he was seeing was a moon of Jupiter.) He developed the idea that it was a body of some kind that was orbiting Jupiter. State this as a hypothesis. And, what testable prediction could this hypothesis imply? That is, what observation would you expect to eventually see if the thing is orbiting Jupiter? (Suppose in his first few observations, and when he makes his hypothesis, so far he has only seen this thing on one side of Jupiter. If it orbited Jupiter and he watched long enough, where else should he see it?)
  2. What other assumptions is Galileo making along with his hypothesis? That is, usually it is not just the case that (H-->P), but rather we assume things like our equipment is working correctly (call that E, so it would be ((H ^ E) --> P), and so on. Can you think of three?
  3. Make up an argument of at least three premises that is valid, where none of the premises is the conclusion alone, and where all the premises are needed to prove the conclusion. Give the argument both in English, and also in our propositional logic. Provide a key so that I can see how your translation works. Prove the argument it in the propositional logic using a syntactic proof (not a truth table).
  4. Make up an argument of at least five premises that is valid, where none of the premises is the conclusion alone, and where all the premises are needed to prove the conclusion. Try to see if you can do it with a conclusion that is a conditional, so that your proof will be a conditional derivation. Give the argument both in English, and also in our propositional logic. Provide a translation key. Prove the argument in the propositional logic using a syntactic proof (not a truth table).

November 9
Due at the beginning of class, translations using predicates and terms. Translate the following into our language. Create a single key, aiming to introduce the minimal number of predicates that you need to symbolize all of the sentences. Remember the arity of the predicate should also be minimal. Ignore differences in tense. Remember that we usually translate "but" with an '^'. NO QUANTIFIERS ARE REQUIRED FOR THIS HOMEWORK. See if you can do 8 and 9 with the same predicate -- think about it for a minute.
  1. Harry is German or English.
  2. Pettigrew is English, but either a rat or a human.
  3. Ron sits between Hermione and Harry.
  4. Moody is tall.
  5. Ron and Hermione are not tall.
  6. Moody is taller than Ron.
  7. If Ron does not sit between Moody and Harry then he sits between either Hermione and Harry or Hermione and Pettigrew.
  8. Moody curses Pettigrew.
  9. Pettigrew is cursed by Harry.
  10. If Ron sits between Hermione and Harry, then Harry does not sit between Ron and Hermione.

Redo on last homework is posted in past assignments and is due 16 November.

November 16
Symbolization using quantifiers. Provide one key for the whole homework, and translate the following sentences into quantified logic (FOL).
  1. Something is green.
  2. There are no cubes.
  3. Everything is yellow.
  4. All cubes are yellow.
  5. All tetrahedrons are not green.
  6. Some cubes are not green.
  7. Nothing is a cube.
  8. Nothing green is a cube.
  9. All green cubes are yellow tetrahedra.
  10. Some yellow cubes are either tetrahedra or green.

November 18
Exam 2 in class. Format is identical to the format of exam 1. Content includes: all material on exam 1; translations with predicates and names. Direct, indirect, conditional proofs. Proving biconditionals.

Mean grade was 75. Standard deviation was 11. Range was 35 to 90. All grades out of 90.