PHL111 Valid Reasoning, Past Assignments

Past Assignments
September 4
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 (using one idea standing for "material")? It would be best if you could. Number 9 is tricky. Remember our discussion of "only if."
  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. Alzheimers is a disease of the mind.
  8. If the mind is brain activity, then Alzheimers 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 Alzheimers is a disease of the mind only if Alzheimers is a disease of the brain.
What do we mean by "translation key"? We mean a translation from English to our logical language. 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 back 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 13
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. We'll do that with a direct derivation.

Remember that we have 3 rules:

Modus Ponens
(φ --> ψ)
Modus Tollens
(φ --> ψ)
Double Negation (two forms)


18 September
You may be interested to hear of a different definition of "argument" (occurs around 2:10) than the one we use in this class: .
20 September
Translate the following sentences. Provide a single key that is all and only for your atomic sentences, and which is independent of your homework answers.
  1. Either Spongebob or Patrick live in a pineapple.
  2. Spongebob lives in a pineapple or under a rock, only if he is a bottom dweller.
  3. If Patrick is a seastar, then Patrick has either five or seven legs.
  4. If Spongebob is a sponge, then he has neither legs nor a nervous system.
  5. Spongebob works at the Krusty Krab or he doesn't live in a pineapple.
Here's an extra question: what do you think is the relation between these four sentences: Neither P nor Q. Not P or not Q. Not both P and Q. Not P and not Q. Think it through and if you have an opinion let me know on your homework.

Here's a do-over version of homework 2, for those of your who were asked to re-do it. Call it 2.2:
  1. Premises: ¬S, (¬Q --> S). Show: Q.
  2. Premises: (S --> Q), (P --> S), ¬Q. Show: ¬P.
  3. Premises: (¬T --> P), (T --> S), ¬P. Show: S.
  4. Premises: P, Q, R, (P --> (R --> (Q-->S))). Show: S.
  5. Premises: (Q --> (R --> S)), (R --> ¬(R --> S)), R. Show: ¬Q.
27 September
Homework! For each of the following problems (1) make a translation key that is only atomic sentences; (2) translate the passage; (3) figure out what the conclusion is and then prove the argument. NOTE! Problems 2 will require you to do a conditional derivation!
  1. Either the Professor Plum or Miss Scarlet killed Colonel Mustard. If Professor Plum killed Colonel Mustard, then Professor Plum was in the kitchen. If Miss Scarlet killed Colonel Mustard, then she was in the drawing room. If Miss Scarlet was in the drawing room, then she was wearing boots. But Miss Scarlet was not wearing boots. So, Professor Plum killed the Colonel.
  2. Either Mrs. White or Mrs. Peacock stole the diamonds. If Mrs. Peacock stole the diamonds, then she was in the billiards room. But if Mrs. Peacock was in the library, then she was not in the billiards room. Therefore, if Mrs. Peacock was in the library, Mrs. White stole the diamond.
A few of you need extra practice with translations. So here's a do over for our last homework.
  1. Neither Fry nor Bender eat Bachelor Chow.
  2. If either Fry or Bender eat Bachelor Chow, then Fry will drink Slurm.
  3. Bender eats bachelor chow only if Fry drinks slurm.
  4. Bender eats bachelor chow if Fry drinks slurm.
  5. Either Bender doesn't eat Bachelor Chow or Fry does.
October 7
A note about office hours. I have a conflict today, and my office hours from 11:30 to 12:30 will have to run instead from 11:30 to 12:00. However, I will be available from 1:30 to 3:00, so hopefully if you need to see me that allows for plenty of time.

Homework Complete the following problems for homework; due at the beginning of class. Problems 1-4 require you to complete a proof. Problems 5-8 require that you make a truth table of a complex sentence to determine when that sentence is true or false.
  1. Premises: (P ^ Q), (P --> R), (Q --> S). Conclusion: (R ^ S).
  2. Premises: (P ^ ¬R), (Q v R), (P <--> S) Conclusion: (S ^ Q).
  3. Conclusion: (((P --> Q) ^ P) --> Q)
  4. Conclusion: ((P v Q) --> (¬P --> Q))
  5. ¬(P --> Q)
  6. (¬P --> Q)
  7. (¬P --> ¬Q)
  8. (P --> ¬Q)
  9. Extra-credit: Is this a valid argument? Premises: (P v Q), (P --> R), (Q --> R). Conclusion: R. If it is valid, how would you prove it?

11 October
Exam 1. In class. Topics include: meaning of each of the connectives; direct and conditional derivations; translations; meaning of valid, sound, tautology, theorem, contingent sentence, and contradictory sentence.

18 October
Here's a homework that will let us review old and new stuff. For problems 1 and 2, do a conditional derivation. For problems 3 and 4, do an indirect derivation. For problems 5 and 6, make a truth table to show when the sentence is true and false, and also explain what kind of sentence it is (tautology, contradictory sentence, or contingent sentence). Problem 7 is the hardest we've ever seen. Try it.
  1. Premises: (P --> Q), (R --> S). Conclusion: ((P ^ R) --> (Q ^ S)).
  2. Premises: (¬S v T), (P --> ¬T), (P v R). Conclusion: (S --> R).
  3. Premises: (P <--> (R v S)), ¬R, ¬S. Conclusion: ¬P.
  4. Conclusion: ¬(P ^ ¬P)
  5. Make a truth table for: (¬P --> (P --> Q)) Is this sentence a tautology, contradictory, or contingent?
  6. Make a truth table for: ((P v Q) ^ ¬(P ^ Q)) Is this sentence a tautology, contradictory, or contingent?
  7. Very challenging: prove the following. Conclusion: (¬(P ^ Q) <--> (¬P v ¬Q))
Hints for 7. (1) This direction is the hard one: (¬(P ^ Q) --> (¬P v ¬Q)). That's proven with a CD, of course, but then try ID for showing (¬P v ¬Q). (2) For your contradiction for that ID, try to make (P ^ Q). You'll need the two parts. So: try to prove each with an ID!

By the way, to make your proofs look prettier, we can have the following trivial rule: repetition. That just means you can repeat any line you like whenever you like -- as long as that line is not in a subproof that you finished! We'd write it out like:
That's handy to make your ID contradictions look nice and clean, since you usually contradict some sentence farther above in your proof.

25 October
Having finished an exam, this is a good time for you to reflect on your study habits. Please watch all five of the videos here. They are short and they can do you a ton of good.

Good news! The recorded lectures are up on Angel! Log in to Angel (try this link) and click on a date. They're pretty poor, but at least they'll allow you do review a lecture.

Homework! For problems 1-8, translate from English into our new logical language. Provide a single translation key. Use names and predicates so that we can see into the structure of the atomic sentences (your key should have no atomic sentences; also, your predicates should have the lowest possible arity). For problems 9-12, we will return to our propositional logic and practice some more proofs. (NOTE: I'm going to have to start grading on proper use of the Fitch bars. I hate to be picky, but I'm starting to see people use lines in their proof that should be unavailable because those lines are in a completed subproof. If you don't put the Fitch bars correctly, then you'll have real trouble seeing what you can and cannot use in your proof.)
  1. Spongebob is a poriferan.
  2. Spongebob is male.
  3. Spongebob is a male poriferan.
  4. Spongebob is not a male poriferan.
  5. Spongebob is either a poriferan or a cniderian.
  6. Patrick is not both a poriferan and a cniderian.
  7. Patrick is not a poriferan, though he is male.
  8. Patrick and Spongebob are male.
  9. Premises: ((P --> Q) ^ (P --> R)) Conclusion: (P --> (Q ^ R))
  10. Premises: (Q v T), (S --> ¬Q), (S --> ¬T) Conclusion: ¬S
    (oops, this proof had a typo before!)
  11. Conclusion: (¬(P --> Q) --> ¬Q)
  12. Conclusion: ((P --> ¬Q) <--> (Q --> ¬P))
A hint for problem 11. You'll need to do an indirect derivation (ID) inside a conditional derivation (CD). For the ID, try to contradict your assumption for conditional derivation (your ACD).

4 November
Translate the sentences from English to the best translation of our first order logic. Provide a key that has the minimal number of predicates, with the minimal arity.
  1. No fish are cats.
  2. Some cats are male.
  3. Some cats are not fish.
  4. Some cats are female.
  5. Some cats are black females.
  6. All cats are mammals.
  7. All black cats are mammals.
  8. No cats are black fish.
  9. Everything is a fish.
  10. Nothing is a fish.
Note you'll not have any names in your key! Also, note that I've posted the preliminary version of the compact description of our language here.

8 November
Homework. Using UI. Complete the following proofs.
  1. Premises: ∀x(Fx --> Gx), Fa, Fb. Conclusion: (Ga ^ Gb).
  2. Premises: ∀x(Hx <--> Fx), ¬Fc. Conclusion: ¬Hc.
  3. Premises: ∀x(Gx v Hx), ¬Hb. Conclusion: Gb.
  4. Premises: ∀x(Fx --> Gx), ∀x(Gx --> Hx). Conclusion: (Fa --> Ha).
  5. Premises: ∀x(Gx v Ix), ∀x(Gx --> Jx), ∀x(Ix --> Jx). Conclusion: Jb.

11 November
Here's an optional homework that can be averaged with the last one if you did poorly on that. Translate the following. Use a key with the minimal number of predicates, each of the minimal necessary arity.
  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.
If you get it in Monday, I'll return it Tuesday so you can use it to help study.

13 November
Exam 2. Material includes all the material on exam 1, plus: indirect proofs, translating phrases from English using the eight forms, and doing proofs that make use of UI.

20 November
Homework. This is a mix of things. For problems 1-3, do a proof. You'll need all your rules now! For problems 4-6, use a truth table to show the argument is invalid. Draw a circle around the row(s) of the truth table that show the argument is invalid. For problems 7-10, make an informal model to show the argument is invalid.
  1. Premises: ∀x(Fx --> Gx), ∃xFx. Conclusion: ∃xGx
  2. Conclusion: (∃x(¬Hx ^ ¬Jx) --> ∃x¬(Hx v Jx))
  3. Premises: ∃x(Hx v Ix), ∀x(Hx --> Jx), ∀x(Ix --> Jx). Conclusion: ∃xJx
  4. Premises: (P --> Q), Q. Conclusion: P.
  5. Premises: (P v Q), ¬P. Conclusion ¬ Q.
  6. Premises: ¬(P ^ Q). Conclusion: ¬(P v Q).
  7. Premises: ∀x(Fx --> Gx), Gb. Conclusion: Fb.
  8. Premises: ∀x(Hx <--> Jx), ∃xJx. Conclusion: Hc.
  9. Premises: ∃x(Jx ^ Kx), ¬Je. Conclusion: ¬Ke.
  10. Premises: ∃x(Jx ^ Kx), Jd. Conclusion: Kd.
As we reviewed in class, the way to make an informal model is, for each argument: to give an interpretation for the predicates and for the names in the argument, such that all the premises of the argument are obviously true and the conclusion is obviously false.

25 November
We'll do 2 things for the next homework: practice UD, and try for the first time to make arguments guided by our logics. This is for some strange reason a difficult thing for us humans to do. We're going to try to use the logic to create valid arguments in English. So, I'm going to ask you to produce a valid argument in natural, normal English. It must have a form that sounds normal -- not logic or weird English-logic mix.

For problems 1-3, complete the proof. These are our first proofs using Universal Deriviation.

For problems 4-8, write a brief argument in proper, informal English. The object here is not to do logic, but to use the logic. You won't prove anything or symbolize anything; rather, you're just showing you can write something in English that, were it translated over to our logic, would be provably valid.
  1. Premises: ∀x(Fx → Gx), ∀x(Gx → Hx). Conclusion: ∀x(Fx → Hx).
  2. Premises: ∀x(Fx v Gx), ∀x(Fx → Hx), ∀x(Gx → Jx). Conclusion: ∀x(Hx v Jx).

    HINT: Do UD to prove a universal. But then what? Consider an ID that assumes ¬(Hx v Jx). Given the premises above, ¬(Hx v Jx) must be false, but it will take some cleverness to show that. I propose that you do another ID to show Hx, or one to show Jx (either would work, because you'll be able to do Add on either one). So, try: UD to show ∀x(Hx v Jx); and inside that UD do an ID to show (Hx v Jx); and inside that ID do an ID to show Hx or to show Jx.

  3. Conclusion: ∀x(Fx v ¬Fx).
  4. Write an argument in English (not in logic!) that is valid, that has at least two premises, and for which at least one of the premises is a conditional.
  5. Write an argument in English (not in logic!) that is valid, that has at least two premises, and for which at least one of the premises is a disjunction (an "or").
  6. Write an argument in English (not in logic!) that is valid, that has at least two premises, and for which at least one of the premises is an existential claim.
  7. Write an argument in English (not in logic!) that is valid, that has at least two premises, and for which at least one of the premises is a universal claim.

4 December
Translate 1 - 5 into our logical language. Use functions to do so (no credit if there are no functions in the translations). (NOTE: you can do the translations of 1-5 using just two functions and four predicates. Try to do it that way.) Prove 6-8.
  1. Tom's father is American.
  2. Tom's mother is Canadian.
  3. Tom's paternal grandfather is Italian.
  4. Tom's maternal grandfather is French.
  5. Both Tom's maternal and his paternal grandmother are American.
  6. Premises: ∀x(Fx ↔ Hx), ∀x(Jx ↔ Hx). Conclusion: ∀x(Fx ↔ Jx).
  7. Premises: ∃x(Fx ^ Hx), ∀x(Fx → ¬Gx), ∀x(Gx v Jx). Conclusion: ∃x(Jx ^ Hx).
  8. Conclusion: (∀x(¬Fx v Hx) → ∀x(Fx → Hx))
Hint for translations. What is a paternal grandmother? It's your father's mother.... Notice what kind of functions you need to capture that notion.

You should not feel you need tutoring in this class (at least, not yet). You can come see me, and I think it's hopefully so far clear-ish. But, there is a good logic tutor available. Starting next week, walk-in tutoring will be from 2-4 in Mahar (the OLS desk attendant will direct students to the room where the tutor will sit). Also, if students would like one on small group tutoring during the week, they can go to and follow the steps to request Andy Buchman via tutortrac. For other qeustions (not regarding scheduling) you can write Andy at "". There to do seem problems with tutortrac -- it logs some students in as tutors, and then they can't see the scheduling options. If this happens to you, unfortunately you have to go to OLS and tell them to fix it for you. You can contact OLS at