Phil 110, Spring 2007 / April 23, 2007 Practice Questions for the Final Exam on May 7, 2007 (9:00am 12:00noon)

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1 Phil 110, Spring 2007 / April 23, 2007 Practice Questions for the Final Exam on May 7, 2007 (9:00am 12:00noon) (1) Which of the following are well-formed sentences of FOL? (Draw a circle around the numbers of the formulas that are well-formed sentences.) 1. Cube(Small(x)) 2. x Cube(Small(x)) 3. Cube(a) 4. x Cube(a) 5. (a c) 6. a = c 7. a Cube(a) 8. Larger(a, a) 9. Larger(a, b) x Cube(x) 10. x Between(x, x, x) 11. x Between(b, x, x) 12. x Between(b, x, c) 13. x (Between(b, x, c) y Small(y)) 14. x (Between(b, x, c) x Small(x)) 15. Tet(a) = Tet(a) 16. (Tet(a) = Tet(a)) 17. Small(Cube(a)) 18. Cube(a) 19. Cube(a) 20. x Cube(x) 21. x x Cube(x) 22. x Cube(x) y Adjoins(x, y) 23. x Cube(x) y Adjoins(a, y) 24. x (Cube(x) y Adjoins(x, y)) 1

2 (2) What are the truth-functional forms of the following sentences? (Use capital letters A, B, C, etc., to represent atomic sentences.) 1. x P(x) x ( Q(x) R(x)) t-f form: 2. x (x = x) t-f form: 3. (( x Q(x) R(a)) x R(x)) x Q(x) t-f form: 4. x (P(x) Q(x)) t-f form: 5. x (Q(x) x (R(x) x S(x))) t-f form: (3) Identify the main connective in each of the following sentences. (Draw a circle around the main connective.) 1. Cube(a) ( Tet(a) Small(a)) 2. (a = c) 3. Cube(a) (( Tet(a) Small(a)) Small(b)) 4. (Cube(a) Cube(c)) 5. Tet(a) Small(a) Medium(b) For the last one, you have to add parentheses first; and there are two acceptable ways to do that. The LPL textbook (p.101) adopts the policy of picking the as the main connective in sequences of conjunctions with no parentheses. (4) Draw a circle around the preceding T or F to indicate whether the respective sentence is true or false: T F 1. All invalid arguments have at least one false premise. T F 2. All tautologies are FO validities. T F 3. All FO validities are tautologies. T F 4. All tautologies are logical necessities. T F 5. All logical necessities are tautologies. T F 6. The symbol is a symbol in FOL. T F 7. Tet(b) is a literal. T F 8. Tet(b) is a literal. T F 9. Tet(b) is a literal. T F 10. Tet(b) is a literal. T F 11. x Tet(x) is a literal. T F 12. Tet(x) is a literal. T F 13. In FOL, an object can have at most one name. 2

3 T F 14. Tet(b) Cube(b) (Tet(b) Cube(b)). T F 15. x Tet(x) x Tet(x). T F 16. Only leopards have spots is logically equivalent to All spotted things are leopards. T F 17. x (Leopard(x) Spotted(x) is a correct translation of All leopards are spotted. T F 18. If P Q, then: Q is a logical consequence of P and vice versa. T F 19. Table(d) Chair(d) says that d is neither a table nor a chair. T F 20. Table(d) Chair(d) says that d is neither a table nor a chair. T F 21. Table(d) Chair(d) says that d is either not a table or not a chair. T F 22. Table(d) Chair(d) says that d is both not a table and not a chair. T F 23. (Table(d) Chair(d)) says that d is neither a table nor a chair. T F 24. (Table(d) Chair(d)) says that d is neither a table nor a chair. T F 25. A valid argument may have false premises. T F 26. A sound argument may have false premises (in a world where it is sound). T F 27. An argument is valid if its conclusion is true whenever its premises are true. T F 28. No valid argument has a false conclusion. T F 29. A sound argument has a true conclusion (in a world where it is sound). T F 30. Formal proofs are as a rule more rigorous than informal proofs. T F 31. An argument may be valid though its conclusion is not actually a logical consequence of its premises. T F 32. Counterexamples may demonstrate invalidity, but examples cannot demonstrate validity. T F 33. Two sentences may have the same truth conditions but not be logically equivalent. T F 34. Two sentences may be logically equivalent but not tautologically equivalent. T F 35. All pairs of sentences that are logically equivalent are tautologically equivalent. T F 36. In a world with no small things, z (Tet(x) Small(x)) would be vacuously true. 3

4 (5) Draw a world which shows that the following argument is invalid: 1. x Square(x) x Striped(x) 2. x (Square(x) Striped(x)) (6) Draw a world which shows that the following argument is invalid: 1. x Square(x) 2. x Square(x) (7) Complete the following truth table (and put an asterisk over the main connective of the given sentence): A B ((A B) (A (B A))) t t t f f t f f Is the given sentence a tautology? yes no Is the given sentence a TT-possibility? yes no 4

5 (8) Complete the following truth table (and put an asterisk over the main connective of the given sentence): A B C ((A B) (A (C B))) t t t t t f t f t t f f Is the given sentence a tautology? yes no Is the given sentence a TT-possibility? yes no (9) Translate the following English sentences into the blocks language: 1. Something is to the left of d, or not. 2. There are no small tetrahedra. 3. Neither b nor d are cubes left of something. 4. Neither b nor d are cubes left of something. [another plausible translation not equivalent to the first one] 5. All cubes and dodecahedra are in the same row but not the same column. 6. All cubes and dodecahedra are in the same row but not the same column. [another plausible translation not equivalent to the first one] 5

6 (10) Fill in the following tables with a Yes or a No in each cell indicating whether the given sentence matches the respective column heading: x Tet(x) x Tet(x) x Tet(x) y Tet(y) x (Tet(x) Tet(x)) x (LeftOf(x, b) RightOf(x, b)) x (Small(x) Medium(x)) ( x Tet(x) x Cube(x) x Dodec(x)) TW Logical FO Necessity Necessity Necessity Tautology (11) Complete the following joint truth table: A B ( A B) A (A B) A B t t t f f t f f Is the third sentence a tautological consequence of the others? yes no If no, draw a star next to a row that serves as a counterexample. If yes, explain why: Is the second sentence a tautological consequence of the others? yes no If no, put an arrow next to a row that serves as a counterexample. If yes, explain why: Which if any of the four sentences are tautologically equivalent? 6

7 (12) Give an example of two sentences that are FO equivalent but which are not tautologically equivalent. (13) Give an example of two sentences that are TW equivalent but which are not FO equivalent. (14) Give an example of two sentences that are TW equivalent but which are not logically equivalent. (15) Give an example of two sentences that are logically equivalent but which are not tautologically equivalent. (16) Give a formal proof showing that x (x = b) is a FO validity: x (x = b) (17) Fill in the missing justifications in the following formal proof: 1. x Cube(x) 2. Cube(b) 3. x Cube(x) Cube(b) 6. x Cube(x) 7

8 (18) Fill in the missing steps in the following formal proof. (Hint: work backwards.) 1. x Cube(x) Elim: 2 4. Elim: 1 5. Intro: 3,4 6. x Cube(x) Intro: 2 5 (19) Fill in the missing justifications in the following formal proof. 1. x (Cube(x) RightOf(x, a)) 2. x (Small(x) Cube(x)) 3. c Small(c) Cube(c) 4. Cube(c) 5. Cube(c) RightOf(c, a) 6. RightOf(c, a) 7. x RightOf(x, a) 8. x RightOf(x, a) (20) Fill in the missing steps in the following formal proof. (Hint: work backwards.) 1. x (Cube(x) Small(x)) 2. x Cube(x) 3. c 4. Elim: Intro: 5 7. Intro: 2, 6 8. Intro: Elim: Elim: 4, x Small(x) Intro:

9 (21) Classify the following statements by placing their numbers in the relevant regions in the Euler diagram below: 1. x ( Tet(x) Cube(x) Dodec(x)) 2. x ( Tet(x) (Cube(x) Dodec(x))) 3. x (Tet(x) Tet(x)) 4. x Tet(x) x Tet(x) 5. ( x Tet(x) x Cube(x)) ( x Cube(x) x Tet(x)) 6. x (Tet(x) Cube(b) x = b) 7. x (Tet(x) Tet(b) x = b) TW Necessities Logical Truths FO Validities Tautologies 9

10 (22) Determine whether the following sentences are true or false in the Tarski-like world depicted below (in 2-D view, where the bottom is to the front). Circle the T or F as appropriate for each sentence. T F 1. z (Tet(z) LeftOf(c, z) T F 2. z (Tet(z) LeftOf(c, z) T F 3. z Cube(z) T F 4. z Cube(z) T F 5. z (FrontOf(z, b) BackOf(z, b)) T F 6. z FrontOf(z, b) z BackOf(z, b) T F 7. z FrontOf(z, b) z (Between(z, b, d) FrontOf(c, z)) T F 8. z FrontOf(z, b) z (Between(b, d, z) FrontOf(c, z)) a c b d 10