DO NOT FORGET!!! WRITE YOUR NAME HERE. Philosophy 109 Final Exam, Spring 2007 (80 points total) ~ ( x)
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1 Page 1 of 13 DO NOT FORGET!!! WRITE YOUR NAME HERE Philosophy 109 Final Exam, Spring 2007 (80 points total) ~ ( x) Part I: Symbolization into RL: Provide symbolizations of the following sentences, using the abbreviations provided below. Bx -- x is a boy. Gx -- x is a girl. Sxy x smiles at y. Wxy -- x winks at y. a = Amy b = Bob 1. Bob winks at Amy only if he winks at every girl. a. (x)(gx Wbx) Wba b. Wab (x)(gx Wbx) c. Wba (x)(gx Wbx) d. (x)(gx Wxb) Wab e. Wba (x)(gx Wxb) 2. Everyone smiles at someone other than themselves. a. (x) ( y) (Sxy x y) b. (x) (y) (x y Sxy) c. ( y) (x) (Sxy x y) d. ( y) (x) (x y Sxy ) e. (x) ( y) (Sxy x y) 3. No boy is winked at by any girl. a. ~ (x) (Bx ( y)(gy Wyx)) b. (x) (Bx ~( y)(gy Wxy)) c. ~ ( x) (Bx ( y)(gy Wyx)) d. ~ (x) (Bx ( y)(gy Wxy)) e. (x) (Bx ~( y)(gy Wyx)) 4. Not all boys smile at girls. a. ~( x) (Gx (y)(by Syx)) b. ( x) (Gx ~ (y)(by Sxy)) c. ( x) (Gx ~ (y)(by Syx)) d. ~(y)(by ( x) (Gx Syx)) e. ~(y)(by ( x) (Gx Sxy)) 5. Some boy winks at every girl who smiles at him. a. ( x)(bx (y)((gy Syx) Wxy)) b. ( x)(bx (y)((gy Sxy) Wyx)) c. ( x)(bx (( y)(gy Syx) Wxy)) d. (y) ( x)( (Bx (Gy Syx)) Wxy)) e. ( x)bx (y)((gy Syx) Wxy) 6. If a girl winks at a boy, then that boy will smile back at her. a. ( x) ( y)((gx By) Wxy) Syx) b. (x)(y) (((Gx By) Wxy) Syx) c. ( x) ( y)((gx By) Wxy) ( x) ( y )Syx d. ( x) ( y)((gx By) Wxy) Sxy)
2 Page 2 of 13 e. ( x)((gx ( y)by) Wxy) Syx) 7. No girl except Amy winks at Bob. a. (x) (Gx ~Wxb) (Ga Wab) b. (x) ((Gx x a) ~Wxb) c. (x) ((Gx x a) ~Wxb) (Ga Wab) d. (x) ((Gx x a) Wxb) e. (x) ((Gx x a) Wxb) (Ga ~Wab) 8. There is a girl who winks at any girl who doesn t wink at her. a. ( x)(gx ( y)((gy ~Wyx) Wxy) b. ( x)(gx ( y)((gy ~Wyx) Wxy) c. (y) ( x) (Gx ((Gy ~Wyx) Wxy) d. ( x) (y) (Gx ((Gy ~Wyx) Wxy) e. ( x)(gx (y)((gy ~Wyx) Wxy) 9. At least two boys have winked at Tom. a. ( x) ( y) ((Bx Wxt) (By Wyt)) b. ( x) ( y) (((Bx Wxt) (By Wyt)) x y) c. ( x) ( y) ((((Bx Wxt) (By Wyt)) x y) (z)((bz Wzt) (z=x z=y))) d. (x)(y)(z) ( ((Bx Wxt) (By Wyt)) (Bz Wzt)) ((x=y y=z) x=z)) e. ( x) ( y) (((Bx Wxt) (By Wyt)) (z)((bz Wzt) (z=x z=y))) 10. The boy who winked at Amy smiled at her as well. a. ( x) ((Bx Wxa) Sxa) b. ( x)( ((Bx Wxa) (y)((by Wya) x=y)) Sxa) c. ( x)( ((Bx Wxa) (y)((by Wya) x y)) Sxa) d. ( x)( ((Bx Wxa) ~( y)((by Wya) x=y)) Sxa) e. ( x)( ((Bx Wxa) ~( y)(by Wya) ) Sxa) Part II: Translations from PL: Using the same abbreviation scheme, translate the following formulas back into intelligible English. 11. (x) ( y) Wyx a. Everyone winks at someone. b. Someone winks at everyone. c. Someone is winked at by everyone. d. Everyone is winked at by someone. e. Everyone winks at everyone. 12. ~ (x)(gx ( y)syx) a. Not all girls smile at someone. b. Not all girls have someone smiling at them. c. Someone smiles at someone besides girls. d. Someone is smiled at by non-girls. e. Someone is smiled at by not all girls. 13. ( y) ((Gy (x)(bx (Sxy ~Syx)) a. No girl smiles back at any boy who smiles at her. b. There is a girl whom some boy smiles at but she doesn t smile back. c. All boys smile at some girl who doesn t smile back at him. d. There is a girl who doesn t smile at any boy who smiles at her. e. There is a girl whom every boy smiles at, but who doesn t smile at any boy. 14. (y)((gy Wya) Wby) a. Bob winks at every girl who winks at Amy. b. Bob is winked at by every girl who winks at Amy.
3 Page 3 of 13 c. Every girl Amy winks at also winks at Bob. d. If every girl winks at Amy, then Bob winks at everyone. e. Every girl Amy winks at is also winked at by bob. 15. (x)( (Bx x t) Sax) (Bt ~Sat) a. Amy smiles at several boys, but she does not smile at Tom. b. All boys smile at Amy, although Amy does not smile back at Tom. c. Amy smiles at every boy except for Tom. d. Every boy besides Tom smiles at Amy e. There is at least one boy whom Amy smiles at, but Tom is not one of them. Part III: Other Questions: 16. If the premises of an argument form along with the negation of its conclusion are all consistent, then that argument form will be a. Valid and sound. b. Valid but unsound c. Invalid and Sound d. Invalid and unsound e. Cannot be determined from the information given 17. If S logically implies T, then the conditional S Τ will be a. a tautology b. a contradiction c. a contingency d. It might be any of these. e. It ll be none of these. 18. Which of the following is an instantiation of ( x)(y) ( (Fxy Gyx) (z) (Pzy Pzx)) a. (y) ( (Ffy Gyf) (z) (Pzy Pzf)) b. ( x)(y) ( (Fxy Gyx) (Pfy Pfx)) c. ( x) ( (Fxf Gfx) (z) (Pzf Pzx)) d. (y) ( (Ffy Gyf) (z) (Pzy Pzx)) e. None of these. For 19-20, Suppose the following rules apply to the relation R: (x) ~Rxx ; (x) (y) (Rxy ~Ryx); (x) (y) (z) ((Rxy Ryz) Rxz) 19. What, then, may we say about R? a. R is transitive and reflexive, but not symmetric. b. R is reflexive and symmetric, but not transitive. c. R symmetric and transitve, but not reflexive. d. R is transitive, symmetric, and reflexive. e. R is transitive, but neither symmetric nor reflexive. 20. Which of the following, then, could the relation R reasonably stand for? a. The relation one bears to oneself (the identity relation). b. The relation a parent bears to a (natural) child. c. The relation siblings bear to each other. d. The relation an ancestor bears to their descendants. e. The relation cousins bear to one another Part IV (21-32): In the space provided, supply justifications for each line of the following derivation that isn t a premise (1 pt apiece): 1. ~ (y) ( x)( ~Rxy ~Px) Pr. 2. ( y) ~ ( x)( ~Rxy ~Px) (21)
4 Page 4 of ( y)(x)( ~Rxy Px) (22) 4. Flag a (23) 5. ~Pa (24) 6. (x)(~rxd Px) (25) 7. ~Rad Pa (26) 8. ~ ~Rad (27) 9. ( x) ~ ~Rax (28) 10. ~ (x) ~Rax (29) 11. ~Pa ~ (x) ~Rax (30) 12. (y)(~py ~ (x) ~Ryx) (31) 13. (y)( (x) ~Ryx Py) (32) 33. In the space below and to the side, briefly specify what has gone wrong with the following derivation. Please identify precisely where in the derivation a violation to the rules occurs. (3 pts) 1. (x) ( y)(rxy) Premise 2. Flag b FS UG 3. ( y)(rby) UI, 1 4. Rba EI, 3, flag a 5. (x) Rxa UG, 4 6. ( y) (x) (Rxy) EG, 5 Part V (32-38): Problems 34. Use the full truth table method to determine whether the following argument form is valid. Show your work and explain how you arrived at your conclusion. (p (q r)), ~p r / r ~q
5 Page 5 of Use the full truth table method to determine whether the following statement form is a tautology, contingency, or contradiction. Show your work and explain how you arrived at your answer. (p q) (p ~q)
6 Page 6 of Construct a derivation of the following argument: (B ~D) A, A ~C / C (B D)
7 Page 7 of Derive the following theorem: (p q) ((p r) (q r))
8 Page 8 of Construct a derivation of the following argument. Be careful about quantifier scope. Fa (x)hx, ( x)gx, (x)(fa ~Gx) / Ha
9 Page 9 of Construct a derivation of the following argument. ( y)(x)(gxy ~Fx) ; (y)(hyy (Fy Sy)) / (x)( y)(gxy ~Hxx)
10 Page 10 of Derive the following theorem. Watch out for flagging restrictions!! (x)( y)(fx Gxy) (x)(fx ( y)gxy)
11 Page 11 of 13 Part VI : Tree Problems Use the tree method to determine the validity or invalidity of the following two arguments. If the argument is invalid, then use the tree to identify an assignment of truth values that shows the argument to be invalid (5 points apiece) Note that in the final for Fall 2008, we shall have only one tree problem, and it will include predicate logic. Be sure you know the rules for developing quantifier formulas!! In addition, we are also planning to have a derivation involving predicate logic with the identity sign. Be sure that you are familiar with the relevant derivation rules for the identity sign (the rule of self-identity and the rule of identity substitution). 41. Premises: P ~(Q T) ; R (~P S) Conclusion: (R ~T)
12 Page 12 of Premises: (~T P) R ; (S Τ) (~R P) Conclusion: ~P R
13 Page 13 of 13 Extra Credit Problem (Attempt this only after you have completed the rest of the exam!!): Can a relation be both symmetric and anti-reflexive? Show why or why not (provide an example if necessary)
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