Philosophy 2100: Introduction to Logic Final Exam I

[Pages:6]Name: __________________________

Philosophy 2100: Introduction to Logic Final Exam I

Sentence Logic (30 points)

Translations 6 points

Translate the following sentences using the dictionary provided. (2 point each)

P = Pablo buys a cat Q = The Queen declares a holiday R = Rafaela goes to the country

S = Sammie buys a cat. T = The trains run on time. V = Venus goes to the country

1. Either Pablo or Sammie bought a cat.

a) P v S

b) P & S

c) P -> S

d) S -> P

2. The trains won't run on time if the Queen declares a holiday.

a) ~T -> Q

b) ~Q -> ~T

c) Q -> ~T

d) Q ~T

3 . Venus and Rafaela will go to the country unless Sammie buys a cat.

a) S -> (V & R)

b) S v (V & R)

c) (V & R) -> S

d) ~S v (V & R)

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Truth Tables (5 points):

I. Complete the truth table and determine whether the following formula is logically true, logically false, or contingent. (5 points)

(P v Q) -> (P & Q)

P Q

(P v

Q) -> (P & Q)

T T T F F T F F

T T T T F T F F

T T F F T F F T

TT T TF F FF T FF F

a) Logically True b) Logically False c) Logically Contingent

Natural Deductions in Sentence Logic (19 points):

I.. Supply the required justification for the derived steps in the following proof. (The justifications should only involve the 8 rules of implication and the 10 rules of equivalence.) (7 points)

1. (P & Q) v (P & R) Pr

2. P -> (~S & R) Pr

3. (Q & T) -> S Pr

4. P & (Q v R)

___________

5. P

___________

6. ~S & R

___________

7. ~S

___________

8. ~(Q & T)

___________

9. ~Q v ~T

___________

10. Q -> ~T

___________

1 Dist 4 Simp 2, 5 MP 6 Simp 3, 7 MT 8 DM 9 CE

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Name: __________________________

II. Use the first 8 rules of implication and 10 rules of equivalence to derive the conclusion of the following argument. (7 points)

(P v Q) -> R ~(S -> R) ~P

1. (P v Q) -> R 2. ~(S -> R) 3. ~(~S v R) 4. ~~S & ~R 5. ~R 6. ~(P v Q) 7. ~P & ~Q 8. ~P

Pr Pr 2 CE 3 DM 4 Simp 1, 5 MT 6 DM 7 Simp

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III. Derive the conclusion of the following argument using any of the 20 rules. (5 points)

P -> Q ~P -> Q Q

1. P -> Q 2. ~P -> Q 3. ~Q -> ~P 4. ~Q -> Q 5. ~~Q v Q 6. Q v Q 7. Q

Pr Pr 1 Contr 2, 3 HS 4 CE 5 DN 6 Dupl

1. P -> Q 2. ~P -> Q

3. ~Q 4. ~P 5. ~~P 6. ~P & ~~P 7. ~~Q 8. Q

Pr Pr AIP 1, 3 MT 2, 3 MT 4, 5 Conj 3-6 IP 7 DN

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Name: __________________________ Predicate Logic (58 points) Translations 10 points Translate the following sentences using the dictionary provided. (2 points each)

a = Ali b = Barry c = Cecilia

1. Ali and Cecilia like Dobermans.

a) Da v Dc

b) x(Ax & Cx)

Px = x likes Poodles. Sx = x feels secure. Dx = x likes Dobermans.

c) Da -> Dc

d) Da & Dc

2. If Barry likes Poodles, then he feels secure.

a) Pb -> Sb

b) Sb -> Pb

c) Pb & Sb

d) Pb -> xSx

3. No one likes both Poodles and Dobermans. a) x~(Rx & Px) b) ~x(Px & Dx)

c) ~x(Px & Dx)

d) x(~Px -> Dx)

4. Someone likes Dobermans, but it isn't Ali.

a) xDx v ~Da ~Da

b) ~Da -> xDx

c) xDx & ~Da

d) ~xDx &

5. Barry and Cecilia like poodles only if everyone does.

a) xPx -> (Pb & Pc) b) (Pb & Pc) -> xPx c) (Pb & Pc) -> ~xPx d) ~xPx -> (Pb & Pc)

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Natural Deductions in Predicate Logic (48 points):

I. Supply the required justification for the derived steps in the following proof (10 points)

1. xPx -> xQx 2. x~(Qx v Rx) 3. ~(Qa v Ra)

4. ~Qa & ~Ra 5. ~Qa 6. x~Qx 7. ~xQx 8. ~xPx 9. x~Px 10. ~Pb 11. ~Pb v Sb

12. Pb -> Sb

Pr Pr

_ _ _

______

2 EE (flag a) 3 DM 4 Simp 5 EI 6 QE 1, 7 MT 8 QE 9 UE 10 Add 11 CE

II. Derive the conclusions of the following arguments. (13 points each)

x ((Qx v Rx) -> Px) x (Sx & ~Px) x (~Qx & ~Rx)

1. x ((Qx v Rx) -> Px) 2. x (Sx & ~Px) 3. (Qa v Ra) -> Pa 4. Sa & ~Pa 5. ~Pa 6. ~(Qa v Ra) 7. ~Qa & ~Ra 8. x (~Qx & ~Rx)

Pr Pr 1 UE 2 UE 4 Simp 3 MT 6 DM 7 EI

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x(Px -> Qx) ~x~Qx -> ~x ~Rx ~x Px v xRx

1. x(Px -> Qx) 2. ~x~Qx -> ~x ~Rx 3. xQx -> ~x ~Rx 4. xQx -> x Rx

5. x Px 6. Flag a

7. Pa

8. Pa -> Qa

9. Qa 10. x Qx 11. x Rx 12. x Px -> x Rx 13. ~x Px v xRx

Pr Pr 2 QE 3 QE ACP FS UI 5 UE 1 UE 7, 8 MP 6-9 UI 4, 10 MP 5-11 CP 12 CE

Name: __________________________

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III. Derive the following theorem. (12 points)

. x ~Px v xPx

1. x Px 2. x Px -> x Px 3. ~x Px v x Px 4. x ~Px v xPx

1. ~(x ~Px v xPx ) 2. ~x ~Px & ~xPx 3. x Px & ~xPx 4. ~~(x ~Px v xPx ) 5. x ~Px v xPx

ACP 1-1 CP 2 C.E. 3 QE

or

AIP 1 DM 2 QE 1-3 IP 4 DN

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