PHIL 012 - Study Sheet for Exam 2 Part I

1. Which of the following statements are logically equivalent to this sentence: Home(Max)  (Home(Claire)  Happy(Carl))

I. (Home(Max)  Home(Claire))  Happy(Carl) II. (Home(Max)  (Home(Claire)  Happy(Carl) III. (Home(Max)  Home(Claire))  (Home(Max)  Happy(Carl)) a. I b. I & II only c. II & III only d. 1, II, and III e. None of the above

2. Which of the following statements are logically equivalent to this sentence: Home(Max)  Home(Claire)  Home(Max)

I. Home(Max)  Home(Claire) II. Home(Max) ^ Home(Claire) ^ (Happy(Carl)  Happy(Carl)) III. Home(Max)  Home(Claire) a. I b. I & II only c. II & III only d. I, II, and III e. None of the above.

3. Which of the following is a negation normal form of the following sentence: ((A  B)  C a. (A  B)  C b. (A  B)  C c. (A  B)  C d. All of the above e. None of the above

4. Which of the following is a negation normal form of the following sentence: (A  B)  C  ((B  A)  B) a. (A  B)  C  (B  A) b. (A  B)  C c. (A  B)  C  (B  A d. All of the above e. None of the above Translate the following sentences into FOL: Note that being unhappy is the opposite of being happy.

5. Either a is not large or it is in back of b.

6. Neither e nor a is to the right of c and to the left of b.

7. Either Carl is happy and Max is unhappy or it is not the case that either Claire is home and Carl is home or Max is happy.

8. It is not the case that Claire is happy, but Max is at home and Carl is unhappy.

Choose the best statements to describe the following sentences: Assume that all sentences like A, B, and C are atomic sentences. All sentences such as Cube(a) follow the rules of Tarski's world.

9. (A  B)  C a. The sentence is unsatisfiable. b. The sentence is satisfiable. c. The sentence is a tautology. d. The sentence is logically true. e. The sentence is tautological and logically true.

10. (Cube(a)  Cube(b))  Cube(c) a. The sentence is unsatisfiable. b. The sentence is satisfiable. c. The sentence is a tautology. d. The sentence is logically true. e. The sentence is tautological and logically true.

11. a = a a. The sentence is unsatisfiable. b. The sentence is satisfiable. c. The sentence is a tautology. d. The sentence is logically true. e. The sentence is tautological and logically true. 12. Larger(a, b)  Larger(b, c)  Larger(c, a) a. The sentence is unsatisfiable. b. The sentence is satisfiable. c. The sentence is a tautology. d. The sentence is logically true. e. The sentence is tautological and logically true.

For the following questions, use these rules: Conjunction Elimination ( Elim) Disjunction Elimination ( Elim) Conjunction Introduction ( Intro) Disjunction Introduction ( Intro) Negation Elimination ( Elim) Negation Introduction ( Intro) Distribution of  over  Distribution of  over  DeMorgan's Theorems (DeM) Reiteration (Reit)

Consider the following proof:

1. (P  R) 2. (P  R) 3. P 4.P   R 5.(P  R)  (P  R) 6. P 7. R 8. P  R 9. (P  R)  (P  R) 10.R 11.P  R 12.(P  R) 13.(P  R)  (P  R) 14. P  R

13. Lines 3 and 7 are: a. Invalid b. the assumptions for a  Elim c. the assumptions for a  Intro d. the assumptions for two  Intro's e. None of the above.

14. The rule for line 8 is a. Invalid b.  Elim, 2 c. DeM, 1 d.  Elim, 7 e. None of the above. 15. Line 12 is a. Invalid b. Valid c. Unnecessary d.  Intro e. None of the above.

15. This is a proof of a. DeMorgan's Theorem b. Distribution of  over  c.  Intro d. Nothing because it is invalid e. None of the above