First Order Logic: Sample Exercises & Answers


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First Order Logic: Sample Exercises & Answers

Translate the following arguments into First Order Logic & demonstrate their validity:

  1. Only philosophers are logicians, and anyone who’s a philosopher is a thinker. Frege and Russell are logicians. Therefore, Frege & Russell are thinkers. (Px, Lx, Tx, f, r)



  1. Carrots are vegetables and peaches are fruits. There are carrots and peaches in the garden. So there are vegetables and fruits in the garden. (Cx, Px, Vx, Fx, Gx)



  1. Since there are no legumes in the garden, and beans and peas are legumes, there are no beans or peas in the garden. (Lx, Gx, Bx, Px)



  1. A horse is an animal. Therefore, whoever owns a horse owns an animal. (Hx, Ax, Oxy)



  1. Dr. Rogers can cure any person who cannot cure himself. Dr. Rogers is a person. Therefore, Dr. Rogers can cure himself. (Px, Cxy, d)



  1. Anything any person owns can be traded. No person can be traded. Therefore no person owns himself. (Px, Tx, Oxy)



  1. Whoever is a friend of Michael will receive a gift. If Michael has any friends, then Eileen is one of them. Therefore, if Ann is a friend of Michael then Eileen will receive a gift. (Rx, Fxy, m, e, a)



  1. If there are any instructors, then if at least one classroom is available they will be effective. If there are either textbooks or workbooks, there will be instructors and classrooms. Furthermore, all classrooms are available. Therefore, if there are textbooks, then some instructors will be effective. (Ix, Cx, Ax, Ex, Tx, Wx)



  1. Whatever begins to exist has a cause. If the universe has a cause, then there is a God. The universe began to exist. Therefore, there is a God. (Bx, Cx, Gx, u)



  1. Any professional can outplay any amateur. Jones is a professional, but he cannot outplay Meyers. Therefore, Meyers is not an amateur. (Px, Ax, Oxy, j, m)



  1. Some people are friends of every person they know. Every person knows someone. Therefore, at least one person is a friend of someone. (Px, Fxy, Kxy)



  1. O’Brien is a person. O’Brien is smarter than any person in the class. Since no person is smarter than himself, O’Brien must not be in the class. (Px, Cx, Sxy, o)



  1. If there are any police officers, then if there are any robbers, then they will arrest them. If any robbers are arrested by any police officers, they will goto jail. There are some police officers and Mickey is a robber. Therefore, Mickey will goto jail. (Px, Rx, Axy, Jx, m)



  1. If anything is wholly good and can prevent any evil, then it will prevent evil. If anything is all-powerful and anything evil exists, then it can prevent the evil. If God will prevent any evil, then no evil should exist. However, there’s some evil. Therefore either God’s not all-powerful or wholly good. (Gx, Ax, Ex, Cxy, Pxy, g)

Answers:

(1)

1) (x)(LxPx)

2) (x)(PxTx)

3) Lf&Lr /Tf&Tr

4) LfPf UI 1

5) PfTf UI 2

6) LrPr UI 1

7) PrTr UI 2

8) Lf Simp 3

9) Pf MP 4, 8

10) Tf MP 5, 9

11) Lr Simp 3

12) Pr MP 6, 11

13) Tr MP 7, 12

14) Tf&Tr Conj 10, 13

(2)

1) (x)(CxVx) & (x)(PxFx)

2) (x)(Cx&Gx) & (x)(Px&Gx) /(x)(Vx&Gx) & (x)(Fx&Gx)

3) (x)(Cx&Gx) Simp 2

4) (x)(Px&Gx) Simp 2

5) Cc&Gc EI 3

6) Pp&Gp EI 4

7) (x)(CxVx) Simp 1

8) (x)(PxFx) Simp 1

9) CcVc UI 7

10) PpFp UI 8

11) Cc Simp 5

12) Vc MP 9, 11

13) Pp Simp 6

14) Fp MP 10, 13

15) Gc Simp 5

16) Gp Simp 6

17) Vc&Gc Conj 12, 15

18) Fp&Gp Conj 14, 16

19) (x)(Vx&Gx) EG 17

20) (x)(Fx&Gx) EG 18

21) (x)(Vx&Gx) & (x)(Fx&Gx) Conj 19, 20

(3)

1) ~(x)(Lx&Gx)

2) (x)[(Bx V Px)  Lx] /(x)[(Bx V Px)  ~Gx]

3) Bx V Px ACP

4) (Bx V Px)  Lx UI 2

5) Lx MP 3, 4

6) (x)~(Lx&Gx) QN 1

7) ~(Lx&Gx) UI 6

8) ~Lx V ~Gx DM 7

9) ~Gx Elim/DS 7

10) (Bx V Px)  ~Gx CP 3-9

11) (x)[(Bx V Px)  ~Gx] UG 10

(4)

1) (x)(HxAx) /(x)(y)[(Oxy&Hy)  Ay]

2) Oxy&Hy ACP

3) HyAy UI 1

4) Hy Simp 2

5) Ay MP 3, 4

6) (Oxy&Hy)  Ay CP 2-5

7) (y)[(Oxy&Hy)  Ay] UG 6

8) (x)(y)[(Oxy&Hy)  Ay] UG 7

(5)

1) (x)[(Px&~Cxx)Cdx]

2) Pd /Cdd

3) ~Cdd AIP

4) (Pd&~Cdd)Cdd UI 1

5) Pd&~Cdd Conj 2, 3

6) Cdd MP 4, 5

7) Cdd&~Cdd Conj 3, 6

8) Cdd IP 3-7

(6)

1) (x)(y)[(Px&Oxy)Ty]

2) (x)(Px~Tx) /(x)(Px~Oxx)

3) Px ACP

4) (y)[(Px&Oxy)Ty] UI 1

5) (Px&Oxx)Tx UI 4

6) Px~Tx UI 2

7) ~Tx MP 3, 6

8) ~(Px&Oxx) MT 5, 7

9) ~Px V ~Oxx DM 8

10) ~Oxx Elim 3, 9

11) Px~Oxx CP 3-10

12) (x)(Px~Oxx) UG 11

(7)

1) (x)(FxmRx)

2) (x)(FxmFem) /FamRe

3) Fam ACP

4) FamFem UI 2

5) Fem MP 3, 4

6) FemRe UI 1

7) Re MP 5, 6

8) FamRe CP 3-7

(8)

1) (x)(Ix[(y)(Cy&Ay)Ex])

2) [(x)Tx V (x)Wx]  [(x)Ix & (x)Cx]

3) (x)(CxAx) /(x)Tx  (x)(Ix&Ex)

4) (x)Tx ACP

5) (x)Tx V (x)Wx Add 4

6) (x)Ix & (x)Cx MP 2, 5

7) (x)Ix Simp 6

8) Im EI 7

9) Im[(y)(Cy&Ay)Em] UI 1

10) (y)(Cy&Ay)Em MP 8, 9

11) (x)Cx Simp 6

12) Cn EI 11

13) CnAn UI 3

14) An MP 12, 13

15) Cn&An Conj 12, 14

16) (y)(Cy&Ay) EG 15

17) Em MP 10,16

18) Im&Em Conj 8, 17

19) (x)(Ix&Ex) EG 18

20) (x)Tx  (x)(Ix&Ex) CP 4-19

(9)

1) (x)(BxCx)

2) Cu(x)Gx

3) Bu /(x)Gx

4) BuCu UI 1

5) Cu MP 3,4

6) (x)Gx MP 2,5

(10)
1) (x)[Px(y)(AyOxy)]

2) Pj & ~Ojm /~Am

3) Pj(y)(AyOjy) UI 1

4) Pj Simp 2

5) (y)(AyOjy) MP 3,4

6) AmOjm UI 5

7) ~Ojm Simp 2

8) ~Am MT 6,7

(11)
1) (x)(Px&(y)[(Py&Kxy)Fxy])

2) (x)[Px(y)(Py&Kxy)] /(x)(y)[(Px&Py)&Fxy]

3) Pm&(y)[(Py&Kmy)Fmy] EI 1

4) Pm(y)(Py&Kmy) UI 2

5) Pm Simp 3

6) (y)(Py&Kmy) MP 4,5

7) Pd&Kmd EI 6

8) (y)[(Py&Kmy)Fmy] Simp 3

9) (Pd&Kmd)Fmd UI 8

10) Fmd MP 7,9

11) Pd Simp 7

12) Pm&Pd Conj 5,11

13) (Pm&Pd)&Fmd Conj 10,12

14) (y)[(Pm&Py)&Fmy] EG 13

15) (x)(y)[(Px&Py)&Fxy] EG 14

(12)

1) Po

2) (x)[(Px&Cx)Sox]

3) (x)(Px~Sxx) /~Co

4) (Po&Co)Soo UI 2

5) Po~Soo UI 3

6) ~Soo MP 1,5

7) ~(Po&Co) MT 4,7

8) ~Po V ~Co DM 7

9) ~Co Elim 1,8

(13)
1) (x)(Px[(y)(RyAxy)])

2) (x)(y)([(Rx&Py)&Ayx]Jx)

3) (x)Px

4) Rm /Jm

5) Pj EI 4

6) Pj(y)(RyAjy) UI 1

7) Pj(RmAjm) UI 6

8) RmAjm MP 5,7

9) Ajm MP 4,8

10) (y)([(Rm&Py)&Aym]Jm) UI 2

11) [(Rm&Pj)&Ajm]Jm UI 10

12) Rm&Pj Conj 4,5

13) (Rm&Pj)&Ajm Conj 9,12

14) Jm MP 11,13

(14)
1) (x)(y)([Gx&(Ey&Cxy)]Pxy)

2) (x)(y)([Ax&Ey]Cxy)

3) (x)([Pgx&Ex]~Ex)

4) (x)Ex /~Ag V ~Gg

5) ~(~Ag V ~Gg) AIP

6) Ag&Gg DM 5

7) Ee EI 4

8) [Pge&Ee]~Ee UI 3

9) (y)([Gg&(Ey&Cgy)]Pgy) UI 1

10) [Gg&(Ee&Cge)]Pge UI 9

11) (y)([Ag&Ey]Cgy) UI 2

12) [Ag&Ee]Cge UI 11

13) Ag Simp 6

14) Ag&Ee Conj 7,13

15) Cge MP 12,14

16) Ee&Cge Conj 7,15

17) Gg Simp 6

18) Gg&(Ee&Cge) Conj 16,17

19) Pge MP 10,18

20) Pge&Ee Conj 7,19

21) ~Ee MP 8,20

22) Ee&~Ee Conj 7,21

23) ~Ag V ~Gg IP 5-22

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