PHL111 Valid Reasoning, past assignments



Past Assignments
August: Here's your first assignment: send me a digital head shot, via email to delancey@oswego.edu. Name the file after yourself, lastnamefirstname. For example, "SmithJoe.jpg." If you don't have a digital camera or don't know anyone who has one, no foul; it's just to help me learn peoples names, and also to compare your photo to the FBI wanted lists. Thanks! NOTE: I can't know who the photo is of if you send it from your cellphone without naming the file or adding some kind of text message! -- it arrives with an ID like 1313125736

29 August: read sections 1.1 and 1.2, pages 2-13 of K&M&M's Logic.

5 September: without looking in your books, decide what you think the truth table (the semantics) for if... then ... should be.

10 September: 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. Ignore tense. Assume that "Jonsie" refers to only one thing. Remember, your goal is to identify the smallest parts of these sentences that are themselves sentences, and then name those parts in your key, and then provide symbolizations for each sentence your key. Look in your book to see how "provided," "on the condition that," and "only if" are used -- see page 11, especially. These are the sentences:
  1. Jonsie is a mammal.
  2. Jonsie is a cat.
  3. If Jonsie is a cat, then Jonsie is a mammal.
  4. Jonsie is a cat, if Jonsie is a mammal.
  5. Jonsie is not a cat.
  6. Jonsie is not a cat, provided that Jonsie is not a mammal.
  7. Jonsie is a fish.
  8. On the condition that Jonsie is a mammal, then Jonsie is not a fish.
  9. Jonsie is a fish only if Jonsie is a chordate.
  10. Jonsie is not a chordate.

17 September: Finish chapter 1 of Kalish & Montague, and complete the following five proofs. Pay attention to the parentheses!
  1. Premises: ~~V, (V --> Q), (Q --> R), (R --> S). Show: S.
  2. Premises: (P --> R), (T --> P), (S --> T), ~R. Show: ~S.
  3. Premises: ((R --> T) --> (P --> S)), ~(P --> S). Show: ~(R --> T).
  4. Premises: (P --> ~S), S. Show: ~P.
  5. Premises: ~P, ((R --> S) --> P), (~(R --> S) --> Q). Show: Q.
If you feel stumped and wonder, how should I proceed, do the following: first, apply your rules a bit, and see if you get any lines that suggest strategies to you. Next, think backwards. In problem 1, you need S. Where can you get S? One premise contains S as a consequent (second half of a conditional). If you had the first half, R, you could get S using modus ponens. But where could you get R? And so on. I call these the flail and think-backwards methods.

September 22: read chapter 1 of Priest.

29 September: second homework due. Complete the following proofs. The first four can be done with direct proofs; the remaining five require conditional derivation. The last two require you to do a conditional derivation within your conditional derivation; that's tricky, but look in your book for examples.
  1. Premises: (~~P --> S), P. Show: ~~S.
  2. Premises: ~(T --> R), ((P --> V) --> (T --> R)). Show: ~(P --> V).
  3. Premises: (~P --> V), ~V, (P --> Q). Show: Q.
  4. Premises: (~(R --> S) --> V), ((R --> S) --> Q), ~Q. Show: V.
  5. Premises: (P --> R), (R --> Q). Show: (P --> Q).
  6. Premises: (S --> R), (P --> S). Show: (~R --> ~P).
  7. #18 in the book, page 27. Premises: ~Q. Show: ((P --> Q) --> ~P).
  8. #21 in the book, page 27. Premises: (Q --> ~R), (~P --> R). Show: (~P --> ~Q).
  9. Premises: (Q --> ~R), (~S --> P). Show: ((P --> Q) --> (R --> S).
  10. #19 in the book, page 27. Premises: (P --> (Q --> R)), (P --> (R --> S)). Show: (P --> (Q --> S)).

6 October: homework. Translations of "and" and "or" and "if...then...." Make one key for all five problems. For the truth tables, tell me whether the statement is a tautology, a contradictory statement, or a contingent statement (neither tautological nor contradictory). [For translations, keep in mind that "and" and "or" can only join two sentences. You must identify sentences that are equivalent to the ones I've give you.]
  1. If Smith and Jones will go to London, then one or the other will go to Berlin.
  2. Neither Smith nor Jones will go to Berlin. [Hint: think of "neither" as "not either...."]
  3. If Smith will go to Berlin then Jones won't.
  4. Smith or Jones, but not both, will go to London.
  5. If Jones will not go to Paris, then Jones won't not go to London.
  6. Make a truth table for the following sentence: (P v ~P)
  7. Make a truth table for the following sentence: (P ^ ~P)
  8. Make a truth table for the following sentence: (~P --> (P --> Q))
  9. Make a truth table for the following sentence: ((P ^ (P --> Q)) --> Q)
  10. Make a truth table for the following sentence: ((P v Q) ^ ~(P ^ Q))
13 October: exam 1 in class. Semantics of all connectives. Truth tables for complex statements. Defintion of valid, contradictory statement, tautology, theorem. Proofs with negation and conditionals using direct derivations and conditional derivations. Translations with all of the connectives.
Results: the mean was 77, the standard deviation was 20. The high grade was 98 (two people got a 98); six people got in the 90s. The low grades were 29, 40, and 46. Remember that this test is just 10% of your grade; if you didn't do as well as you would like, you have lots and lots of time still to turn your grade around.

20 October: Homework. Complete the following four proofs. You will need to use indirect derivation (although you may find you need it for a subproof, and not for your main proof).
  1. Conclusion: (~Q --> ~(P ^ Q))
  2. Premises: (P ^ R), (P --> S), (Q --> ~(S ^ R)). Conclusion: ~Q. [Hint: the contradiction is not always an atomic sentence. For example, ~(S ^ R) and (S ^ R) contradict each other.]
  3. Conclusion: ((P ^ ~Q) --> ~(P --> Q)). [Hint: always do a conditional derivation to show a conditional; but then what is the second part of your conditional? Might it not be shown using an indirect derivation?]
  4. Conclusion: (~(R --> T) --> ~T). [Hint: super hard, until you see that you can contradict your assumption for conditional derivation.]
24 October: Consider the following two natural language arguments. One is invalid, and the other is valid. For each, provide a single key and translate the argument. Then, determine which is valid. For the valid argument, prove the argument. For the invalid argument, create a truth table that shows the argument is invalid (remember, this means that you'll have a column for each premise, a column for the conclusion, and you will show clearly that there are kinds of situations -- rows -- where all the premises are true but the conclusion is false). 29 October read chapter 2 of Priest, Logic.

October 31: Homework. Complete the first three proofs, and then make a truth table for the last two. You may use, in the proofs, theorem T93 or any lower-numbered theorem if you like (they aren't needed but can save you time) (see pages 109-110 in Montague to see a list of theorems).
  1. Show: (P <--> Q). Premises: (P <--> R), (R <--> S), (S <--> Q).
  2. Show: ((Q v P) ^ T). Premises: (P <--> R), (R ^ ~S), (S v T).
  3. Show: R. Premises: ((T v R) <--> S), (S ^ ~Q), (T <--> Q).
  4. Make the truth table for the following sentence: (~(P v Q) <--> (~P ^ ~Q)).
  5. Make the truth table for the following sentence: (~(P ^ Q) <--> (~P v ~Q)).)).
November 7: no class (I'll be out of town at a conference). But, you can drop off any time during the day on the 7th (or before) the following homework at the philosophy department office right next door to our class. Staple it if you have more than one page. Write very clearly on it your name and also at the very top "FOR DELANCEY" so the secretary knows where to put it. Thanks!
Translations using predicates and terms. Translate the following into our language. Create one key, aiming to introduce the minimal number of predicates that you need to symbolize all of the sentences. Ignore differences in tense. [Hint: there are some arity-2 predicates, and at least one arity-3 predicate in here.]
  1. Harry is French 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 just in case Pettigrew is not sitting between Harry and Ron.
  9. Pettigrew was cursed by Moody or Harry.
  10. If Ron sits between Hermione and Harry, then Harry does not sit between Ron and Hermione.

10 November: read chapter 2 of Priest, Logic. Don't be confused: many logical systems use a backwards E for our \/, and they use an upside down A for our /\. The meanings are still the same. Also, Priest uses prefix notation, he writes the name/term before the predicate. That too is not a substantial difference. What matters is only that one stick with a convention once deciding upon it. We'll stick with post-fix notation, writing a name after a predicate. So, we write Fa to mean "a is an F" and Priest writes aF.

November 12: homework doing translations of expressions with quantifiers. Write a single key and translate the following sentences. The last four demand you think a minute, but I can promise you they are still variations of the four forms we saw.
  1. Something is a cat.
  2. Everything is a fish.
  3. Something is not a cat.
  4. Nothing is a fish.
  5. No fish are cats.
  6. Some cats are fish.
  7. Some cats are not fish.
  8. All fish are cats.
  9. Some black cats are spiny fish.
  10. All spiny green fish are black female cats.
  11. No green cats are black fish.
  12. Some green spiny cats are black female fish.

November 19: exam 2. All of the propositional logic (truth tables; doing proofs with any connective; translations with any connective; meaning of tautology, theorem, contradiction; indirect proofs, conditional proofs, direct proofs). Translating sentences using names and predicates and quantifiers.
Grades: the mean was a 79, the standard deviation was a 13.5. There were two grades of 100. The low grade was a 56.

December 1: homework due. Below are four arguments. Symbolize them, and then prove them. They will require the inference rules EG and UI but they will not require EI or Universal Derivation.
  1. Premises: All Horta are silicon. McCoy is not silicon. Conclusion: Something is not Horta.
  2. Premises: All Parisians eat crepes. All who eat crepes like apricot jam. Smith is Parisian. Conclusion: Something likes apricot jam.
  3. Premises: Nebbiolo is better than Cabernet. Anything better than Cabernet is better than catawba. Conclusion: Something is better than catawba. [Hint: what arity is "better than"? Note then that your quantifier need only grab one slot in a predicate of arity greater than 1. For example, if "Fxyz" is the predicate "x is between y and z," then \/xFxab means someone is between a and b, \/xFaxb means a is between someone and b, and \/xFabx means a is between b and someone. Or, if "Gxy" is the predicate "x is bigger than y," then \/xGax means a is bigger than something, while \/xGxa means something is bigger than a.]
  4. Premises: Patrick is taller than Spongebob. Patrick is dumber than Spongebob. Conclusion: Someone is both taller than and dumber than Spongebob. [Hint: look at the last hint, and note also that this conclusion is of the form, some F are G.]