PHL111 Valid Reasoning, Past Assignments



Past Assignments
August 29
Please read section 1.1 of Logic.
August 31
Translate the following sentences from English into our logical language, the Propositional Logic. Provide one key for the entire assignment that shows to which (hopefully atomic) English sentences your propositional logic sentence letters correspond. Always try to show as much as you can about the structure of the sentence in our logic (that is, don't translate them all as a single letter, like P). Hand in your answers and translation key at the beginning of class. Write your homework neatly, since I might want to show yours on the overhead as an example.

For these problems: The last one is hard! It nests "if... then..."s. Try it. Also, think hard about "material" and "immaterial" -- can you show their relation using our logic? Ideally, you should. Number 9 is tricky. Look in your book for the discussion of "only if." Hopefully also we'll get to this one in class.
  1. The mind is brain activity.
  2. The mind is material.
  3. The mind is immaterial.
  4. If the mind is immaterial, then the mind is not brain activity.
  5. If mind is brain activity, then the mind is material.
  6. Provided the mind is not immaterial, then the mind is brain activity.
  7. Schizophrenia is a disease of the mind.
  8. If the mind is brain activity, then schizophrenia is a disease of the brain.
  9. The mind is brain activity only if the mind is material.
  10. If the mind is brain activity, then schizophrenia is a disease of the mind only if schizophrenia is a disease of the brain.

Some folks have asked me, what do you mean by "translation key"? Well suppose instead one of our sentences was....
8. If Spongebob lives in Bikini Bottom, then Spongebob pays no property taxes.
You might have a key that includes the following:
English ..................................... Propositional Logic
Spongebob lives in Bikini Bottom........... P
Spongebob pays no property taxes........... Q
And then if you translated the sentence:
8. (P --> Q)
Someone would be able to use your key to determine how to translate your logical sentence into an English sentence.

So, for this homework, you'll write instead one key (put it at the top of the page maybe), which will have a column of atomic English sentences, and beside it a column of atomic logic sentences (P, Q, R....).

September 7
Please read sections 1.2 and 1.3 of Logic.

September 14
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. You will need the rules MP, MT, and DN.
  1. Premises: ~Q, (~Q --> S). Show: S.
  2. Premises: (S --> ~Q), (P --> S), ~~P. Show: ~Q.
  3. Premises: (T --> P), (Q --> S), (S --> T), ~P. Show: ~Q.
  4. Premises: R, P, (P --> (R --> Q)). Show: Q.
  5. Premises: ((R --> S) --> Q), ~Q, (~(R --> S) --> R). Show: R.
For this assignment, let's be very precise about the rules, and not make leaps based upon your understanding (that is, only write down in your proof stuff that the rules specifically allow).

Some of you asked me to make some common sense of these sentences. Well, consider problem 3. An English argument that would look like that might be:
If Steve goes to London then he will go to Piccadilly circus. If Steve goes overseas, then he will go to England. If Steve goes to England then he will go to London. But Steve did not go to Piccadilly circus. Therefore, Steve did not go overseas.
That argument might take a little thought, but I bet that if you think about it, you'll agree that the conclusion must be true: Steve did not go overseas. But now imagine a skeptic comes along and says, "Prove it! Prove that Steve did not go overseas!" That's what you're doing in your homework: proving that if the premises are true, the conclusion must be true. And the rules only let us write down stuff that must be true, if the earlier lines in your proof are true, so if we can get the conclusion from the premises and the applications of our rules, then the conclusion must be true.

Or, consider problem 4. An English argument that would look like that might be:
Tom rides a horse. Tom herds cattle. Provided that Tom herds cattle, if he rides a horse then he is a cowboy. We conclude that Tom is a cowboy.
We're going to prove to the skeptic that Tom must be a cowboy.

September 18
Class meets on Tuesday, not Monday, this week only.

A general requirement: I'd like to see each of you individually some time this semester, hopefully before Thanksgiving recess. I'd just like to talk with you for a few minutes to find out how things are going, and perhaps figure out what in logic you want more or less time with, and also maybe to discuss even how logic can be helpful to you in your future studies. Try to schedule 30 minutes with me during my office hours, or if those don't work we can find another time. Use email to set it up.

September 21
Read sections 2.1-2.3 in your book.

Homework! Two parts.

Part 1: Translate the following argument, providing a single key. This is a more informal statement of an argument, and your task is to figure out what the conclusion is, and also to recognize what should be ignored as irrelevant to the argument. Then, after translating the argument, prove the argument using an argument. Note: I tried to make this a bit easier by adding the word "both" in the penultimate sentence.
The Professor killed the Butler. If the professor was in the pantry, then the professor had the candlestick. Provided that the professor did not enter the billiards room, then the professor did not have the candlestick. The butler was from England, and liked billiards. And, the professor killed the butler if the professor both entered the billiards room and had the candlestick. But, the professor was in the pantry.
Part 2: write a truth table for each of the following sentences, with the aim of showing when the sentence is true (in terms of when its parts are true).
  • ~(P --> Q)
  • (~P --> Q)
  • (~P --> ~Q)
  • ~(P ^ Q)
  • (~P ^ ~Q)
  • (P ^ ~P)
  • (~(P --> Q) --> ~Q)
Part And: a reminder that I'd like to see each of you individually some time this semester, hopefully before Thanksgiving recess. I'd just like to talk with you for a few minutes to find out how things are going, and perhaps figure out what in logic you want more or less time with, and also maybe to discuss even how logic can be helpful to you in your future studies. Try to schedule a few minutes with me during my office hours, or if those don't work we can find another time. Use email to set it up.

28 September
Please complete the following proofs. Each will require a conditional derivation. Hint: You'll need a proof within a proof for number 3.
  1. Premises: (P --> R), (~T --> Q), (R --> ~T). Show: (P --> Q)
  2. Premises: (P --> S), (Q --> T). Show: ((P ^ ~T) --> (S ^ ~Q))
  3. Show: ((P --> ~Q) --> (Q --> ~P))
  4. Show: (((P --> Q) ^ P) --> Q)

October 5
Homework due at the beginning of class. Translate each of the following sentences, then write a truth table for each of the sentences, to show when the sentence is true or false. If the sentence is a tautology, prove it using our syntactic proof methods (e.g., using a conditional derivation).
  1. Kyleigh will go to Nagasaki or Tokyo.
  2. If Zach goes to both London and Paris, then Zach goes to Paris.
  3. If Gloria is going to Disneyland or Disneyworld, but it turns out she's not going to Disneyworld, then she is going to Disneyland.
  4. Morgan is going to Boston or New York, but not both.
Here's one other kind of problem, which requires us to use our logic to produce an argument. If you get stumped, write it out with Ps and Qs first, and then fill in the sentence symbols after. Also, why not try to make it about something in your major?
5. Write a valid argument in normal English with at least two premises, one of which is a conditional, and one of which is a negation sentence.

6. Write a valid argument in normal English with at least two premises, one of which is a disjunction (an 'or'), and one of which is a negation sentence.

October 8, 10
This is a good time for you to reflect on your study habits. Before class, please watch all five of the videos here. They are short and they can do you a ton of good. I might give you a short quiz or problem in class, to test whether you learned what Dr. Chu is teaching.
October 10
Read chapter 2 of Kalish and Montague. Homework 5 due at the beginning of class.
  1. Show: (P <--> Q). Premises: (P <--> R), (R <--> S), (S <--> Q).
  2. Show: (P ^ T). Premises: (P <--> R), (R ^ ~S), (S v T).
  3. Show: R. Premises: ((T v R) <--> S), (S ^ ~Q), (T <--> Q).
  4. Show: 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)).)).

October 11
Let's do a few things.
  1. Let's finally talk about what we learned from Dr. Chew. Review the videos, if you have a chance.
  2. We can review a little our language. Here's a practice problem. For the following arguments: translate the argument; make a truth table to show if the argument is valid; and if the argument is valid then prove it using our syntactic proof method.
    • If Jim wins the lottery, then he will go to New York City. If Jim wins the lottery, then he will buy new skis. Jim does not win the lottery. Therefore, Jim will neither go to New York City nor buy new skis.
    • If Bobby does the crime, then he can do the time. Either Bobby can't do the time, or he's dishonest. But Bobby is honest. Therefore, Bobby is an honest man who won't do the crime.
    Partial solutions are here.
  3. Let's remind ourselves of the conceptual stuff. Can you define: valid, sound, tautology, theorem, contradictory sentence, contingent sentence, argument? And: what are the three uses of a truth-table?

October 15
Exam 1.

The mean was 72. The median was 80. The standard deviation was 17. The range was 40 to 93.

October 17
Read chapter 2 of Kalish and Montague. Review of Exam 1. Then: a striking new proof method. Before class, see if you can prove: ((~P v ~Q) --> ~(P ^ Q)) or if you can prove (~(P v Q) <--> (~P ^ ~Q)). Both are tautologies.

October 22
Homework. For problems 1-4, use a truth table to show that the argument is invalid. This means a column for each premise, a column for the conclusion (you may have other columns to help you know how to fill in those columns...), and you identify the kinds of situations (rows) where all the premises are true and the conclusion is false. Recall that this is our third use for truth tables (1. define a connective; 2. determine under what conditions a sentence is true or false; 3. determine if an argument is valid or invalid). For problem 5, use a truth table to show the sentence is not a tautology. What kind of sentence is it? For problems 6 & 7, prove the theorem.
  1. Premises: (P v Q), P. Conclusion: ~Q.
  2. Premises: (~R --> S), R. Conclusion: ~S.
  3. Premises: (T --> V), ~T. Conclusion: ~V.
  4. Premises: ~(R^S). Conclusion: ~R.
  5. Show that this is not a tautology: ((P ^ Q) <--> (P v Q)). What kind of sentence is it?
  6. Prove theorem 43, page 109: (~P --> ~(P ^ Q)). This will require a subproof where you do indirect deriviation.
  7. Prove theorem 63, page 109: ((P^Q) <--> ~(~P v ~Q)) This will require two conditional derivation subproofs and each of those will require a indirect derivation subproof! This is our hardest proof yet!

October 26
Homework, mostly to review.
  1. Make a truth table to prove this argument is invalid. Circle all and only rows that show the argument is invalid. Premises: (P --> Q), (P --> R), ~P. Conclusion: (~R ^ ~Q). Sorry, this will require 8 rows.
  2. Prove theorem T37 page 108: ((P --> Q) <--> ~(P ^ ~Q))
  3. Prove the following argument. Premises: (P v Q), (P --> R), (Q --> R). Conclusion: R.
  4. Consider the following argument in English: "All men are mortal. Socrates is a man. Therefore, Socrates is mortal." First, in your opinion, is the argument valid? Second, if you think it is valid, can you symbolize the argument and then prove it?

31 October
Translations using predicates and terms. Translate the following into our language. Create a key, aiming to introduce the minimal number of predicates that you need to symbolize all of the sentences. Ignore differences in tense.
  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.

5 November
Homework doing translations and using UI. Translate problems 1 through 12. Use one key.
  1. Everything is a spider.
  2. Something is a spider.
  3. Nothing is a spider.
  4. Something is not black.
  5. Some spiders are arthropods.
  6. All spiders are arthropods.
  7. No spiders are arthropods.
  8. Some spiders are not arthropods.
  9. Some black spiders are female arthropods.
  10. No red spiders are male arthropods.
  11. Some male spiders are not red.
  12. All spiders are red male arthropods.

9 November
Homework using UI and EG. Complete the following proofs:
  1. Premises: /\x(Fx --> Gx), /\x(Gx --> Hx), Fa. Conclusion: Ha.
  2. Premises: /\x(Gx <--> ~Ix), Gb, /\x(Ix v Jx). Conclusion: Jb.
  3. Premises: /\xLxd, /\x(Lxd --> ~Hxd). Conclusion: ~Hcd
  4. Premises: Fa, /\x(Fx --> Jx), /\x(Fx --> Ix). Conclusion: \/x(Jx ^ Ix)
  5. Premises: /\x(Fx --> Gx), ~Gb, /\x(Gx <--> Ix) Conclusion: \/x(~Fx ^ ~Ix)
14 November
Exam 2. New material will include: translating using quantifiers; proofs using indirect proof, and using the rules UI and EG.

You asked for practice problems.
  • This one might I think use most of our propositional logic rules. Conclusion: ((P^Q) <--> (RvS)). Premises: (P <--> T), ((~Q v U) ^ (~U v Q)), ((T ^ U) <--> S), (R --> ~(P <--> T)).
  • Translations with higher arities. It's hard to find natural examples in everyday English, but not hard in math. Translate the following. Note that the first is an atomic sentence, so "and" in it is not the same as our connective. Assume your domain of discourse is natural numbers.
    1. 7 and 8 are factors of 56.
    2. 8 and 7 are factors of 56.
    3. 3 and 9 are not factors of 56.
    4. There is a number such that it and 9 are factors of 27.
    5. 3 and some number are factors of 27.
    6. 3 and 9 are factors of some number.
Solutions are here.

19 November
Homework due. You can of course hand this in early if you like; that is always permitted.

Complete the following proofs. The last two are challenging; do your best on them.
  1. Premises: \/x~Fx, /\x(Fx v Gx). Conclusion: \/xGx.
  2. Premises: \/x(Fx ^ Gx), /\x(Hx <--> Gx). Conclusion: \/x(Gx ^ Hx).
  3. Premises: /\yFy, /\x(Fx --> Gx). Conclusion: /\zGz.
  4. Premises: /\y(Fy <--> Gy), /\x(Gx <--> Hx). Conclusion: /\x(Fx <--> Hx).
  5. Theorem T203. Conclusion: (~/\xFx <--> \/x~Fx)
  6. Theorem T204. Conclusion: (~\/xFx <--> /\x~Fx)
Numbers 5 and 6 are super hard. I will give you most of the credit for the problem if you get half of the proof. For each, one half of the biconditional is easier to prove. For 5, (\/x~Fx --> ~/\xFx) is easier; for 6, (/\x~Fx --> ~\/xFx) is easier.

In class, we'll discuss cool shocking discoveries of logic as a holiday celebration.

Regarding the test. The mean was 75, the median was 77, the standard deviation was 14. The highest grades were two tied 94s.

30 November
Arguments 1-4 are valid. Prove them. Arguments 5-8 are invalid. Show that they are invalid using informal models. (Hint: #3 and #4 will probably require an ID subproof. Don't do #3 using disjunctive syllogism or constructive dilemma or a similar rule.)
  1. Conclusion: /\x(Fx --> Gx) --> /\x(~Gx --> ~Fx)
  2. Premises: /\x(Gx ^ Fx), /\x(Fx --> Ix), /\x(Gx --> Jx). Conclusion: /\x(Ix ^ Jx).
  3. Premises: /\x(Gx v Fx), /\x(Fx --> Ix), /\x(Gx --> Ix). Conclusion: /\xIx.
  4. Premises: /\x(Hx v Ix), ~\/xIx. Conclusion: /\xHx.
  5. Premises: /\x(Fx --> Gx), ~Fa. Conclusion: ~Ga.
  6. Premises: \/xFx, \/xGx. Conclusion: \/x(Fx ^ Gx).
  7. Premises: \/x~Fx. Conclusion: \/xFx.
  8. Premises: ~/\xFx. Conclusion: ~\/xFx.
  9. Construct an argument in English that uses at least one existentially quantified phrase, has at least two premises, and is valid. The argument must be precise but written in colloquial English (not in logic-speak).
  10. Construct a different argument in English that uses at least one universally quantified phrase, has at least two premises, and is valid. The argument must be precise but written in colloquial English (not in logic-speak).

3 December
Read Kalish, Montague, and Marr chapter IV.
In class, we'll (1) review the homework; (2) see if you have any questions about the final; (3) continue discussing functions; (4) review the senses of "is"; and (5) see a handy trick you can do with identity. You'll need all this for the homework!

5 December
I've added some problems, because on Friday's homework there was still some deep confusion about universal derivation; and also not much luck with constructing arguments.

Translate sentences 1-12 into our logical language. Remember that "is" is an arity two predicate; you may use "=" if you like. See if you can do #5 without introducing a new function other than the one you used for #2-#4. Assume your domain of discourse is people. Prove the arguments in problems #13 and #14. Do #15.
  1. Steve is older than Tom.
  2. Steve is older than the father of Tom.
  3. The father of Steve is older that the father of Tom.
  4. Somebody is older than Tom.
  5. Somebody is not older than Tom.
  6. Nobody is older than Tom's father.
  7. Everybody is older than Tom's father.
  8. Somebody is Tom. [This is also sometimes considered equivalent to: "Tom exists."]
  9. Steve is Tom's father.
  10. Somebody is Steve's father.
  11. Doug is Tom's paternal grandfather.
  12. Steve is not Tom's paternal grandfather.
  13. Premises: /\x(Fx --> Gx), /\z(Fz --> Hz). Conclusion: /\y(Fy --> (Gy ^ Hy)).
  14. Premises: /\x(Fx --> Gx), (\/xGx --> \/xHx), (\/xHx --> /\x(Ix v Jx)), Fc, /\x~Ix. Conclusion: /\xJx
  15. Write an argument in colloquial English that includes at least two premises, and one of the premises is a universal statement, and the argument is valid. Furthermore, the argument cannot be trivial; that is, the argument must not repeat a premise as its conclusion. Finally, colloquial means colloquial. No symbols. No numbered lists. No PREMISE labels. No "for any x, if x is a G..." logic-speak stuff.

7 December
We'll review the homework, and solve some problems, including some proofs with functions in them.