Section 1 True/False (5 points each)

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Philosophy 240, Introduction to Logic

Exam 3, Practice

Section 1 True/False (5 points each)

1. E is a 2-line rule. 2. E is a 2-line rule. 3. A valid argument may have a false conclusion. 4. A sound argument may be invalid. 5. An invalid argument may have a true conclusion. 6. No sound argument is invalid. 7. RAA is a 1-line rule. 8. I is a 1-line rule.

- False - True - True - False - True - True - False - True

Section 2 Translations (6 points each) Translate each of these sentences using the following translation scheme.

Bx = `x is a book' Hx = `x is a hardback' Px = `x is a paperback' Ex = `x exists' Lxy = `x is longer that y' a = `Logic Primer' b = `Crime and Punishment'

1. Logic Primer is a paperback if Crime and Punishment is a hardback.

Answer: (Hb Pa)

2. Among books, only hardback and paperback exist.

Answer: x(Bx (Ex (Hx v Px)))

3. All books are paperbacks.

Answer: x(Bx Px)

4. Not all books are hardbacks on the condition that paperbacks exist.

Answer: (x(Px & Ex) ~y(By Hy))

5. Either Crime and Punishment is a hardback, or every book is longer than Logic Primer.

Answer: (Hb v x(Bx Lxa))

Section 3 Proofs (6 points each) Prove the following sequents. You may use both primitive and derived rules.

1. x(Fx v Gx), x(Gx Hx), x~Fx xHx

1 2 3 1 3 1,3 2 1,2,3 1,2,3

(1) x(Fx v Gx) (2) x(Gx Hx) (3) x~Fx (4) Fa v Ga (5) ~Fa (6) Ga (7) Ga Ha (8) Ha (9) xHx

A

A A xHx 1E 3 E 4,5 vE 2 E 6,7 E 8 I

2. xPx, x(Qx v Rx), x(~Rx & Sx) x(Px & Qx)

1 2 3 4 3 3 2 2,3 2,3,4 2,3,4 1,2,3

(1) xPx (2) x(Qx v Rx) (3) x(~Rx & Sx) (4) Pa (5) ~Ra & Sa (6) ~Ra (7) Qa v Ra (8) Qa (9) Pa & Qa (10) x(Px & Qx) (11) x(Px & Qx)

A

A A x(Px & Qx) A (for E on 1) 3 E 5 &E 2 E 6,7 vE 4,8 &I 9 I 1,10 E(4)

3. xFx xGx, xHx ~xGx xFx x~Hx

1 2 3 1,3 1,2,3 1,2,3 1,2

(1) xFx xGx (2) xHx ~xGx (3) xFx (4) xGx (5) ~xHx (6) x~Hx (7) xFx x~Hx

A A xFx x~Hx A (for I) 1,3 E 2,4 MTT

5 QE 6 I(3)

4. y(Fy (Fy & Hy)), y((Fy Hy) Gy) yGy

1 2 3 3 5 2 3 2,3 2,3 2,3 2,3 2,3,5 2,3,5 2,5 1,2

(1) y(Fy (Fy & Hy)) (2) y((Fy Hy) Gy) (3) ~yGy (4) y~Gy (5) Fa (Fa & Ha) (6) ((Fa Ha) Ga) (7) ~Ga (8) ~(Fa Ha) (9) Fa & ~Ha (10) Fa (11) ~Ha

(12) Fa & Ha (13) Ha (14) yGy (15) yGy

A A yGy A (for RAA) 3 QE A (for E on 1) 2 E 4 E 6,7 MTT 8 NEG 9 &E 9 &E 5,10 E 12 &E 11,13 RAA (3) 1,14 E(5)

5. x(Px (Qx & Rx)), xPx x~Rx |- ~xPx

1 2 3 4 1 1,4 2,3 2,3 1,4 1,2,4 1,2,3 1,2

(1) x(Px (Qx & Rx)) (2) xPx x~Rx (3) xPx (4) Pa (5) Pa (Qa & Ra) (6) Qa & Ra (7) x~Rx (8) ~Ra (9) Ra (10) ~xPx (11) ~xPx (12) ~xPx

A A A (for RAA) A (for E on 3) 1 E 4,5 E 2,3 E 7 E 6 &E 8,9 RAA (3) 3,10 E (4) 3,11 RAA (3)

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