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Chapter 08 Test A
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INSTRUCTIONS: Select the correct translation for each statement.
1. If either Ann or Charlie wins the lottery, then George will celebrate.
a. (x)(Ax ∨Cx) ⊃ Gx
b. (Wa ∨Wc) ⊃ Cg
c. (Wa • Wc) ⊃ Cg
d. (∃x)[(Ax ∨ Cx) • Gx]
e. (x)[(Wa ∨Wc) ⊃Cg]
2. There is a lamp in the bedroom.
a. (∃x)(Lx ⊃ Bx)
b. (x)(Lx ⊃Bx)
c. (x)(Lx • Bx)
d. (∃x)(Lx • Bx)
e. (∃x)Lx • (∃x)Bx
3. A raccoon is not a mongoose.
a. (x)(Rx ⊃ ∼Mx)
b. ∼(x)Rx ⊃ Mx)
c. (x)Rx ⊃ ∼(x)Mx
d. (∃x)(Rx • ∼Mx)
e. Rx • ∼Mx
4. A small bird landed on the roof.
a. (x)[Sx ⊃(Bx ⊃Lx)]
b. (x)[(Sx • Bx) ⊃ Lx]
c. (∃x)(Sx • Bx) • (∃x)Lx
d. (∃x)[(Sx • Bx) • (y) Ly]
e. (∃x)[(Sx • Bx) • Lx]
5. Not every model is emaciated.
a. (x)(Mx ⊃ ∼Ex)
b. (x)∼(Mx ⊃ Ex)
c. (∃x)(Mx • ∼Ex)
d. ∼(∃x)(Mx • Ex)
e. (∃x)(Mx • Ex)
6. Every book in the library is misplaced or checked out.
a. (x)[(Bx • Lx) ⊃ (Mx ∨ Cx)]
b. (x)[(Bx ⊃Mx) • (Lx ⊃ Cx)]
c. (x)[(Bx ⊃Lx) ⊃(Mx ∨Cx)]
d. (x)[(Bx • Lx) • (Mx ∨ Cx)]
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e. (x)[(Bx ∨Lx) ⊃(Mx • Cx)]
7. None but the experienced drivers are cautious and safe.
a. (x){Dx ⊃[Ex ⊃(Cx • Sx)]}
b. (x)[(Ex • Dx) ⊃ (Cx • Sx)]
c. (x)[(Cx • Sx) ⊃ (Ex • Dx)]
d. (x){Dx ⊃[(Cx • Sx) ⊃ Ex]}
e. (∃x)[(Dx • Cx) ⊃ (Ex • Sx)]
8. Goats and sheep are contented only if they are not hungry.
a. (x){[(Gx • Sx) ⊃ (∼Hx ⊃ Cx)]
b. (x){[(Gx ∨Sx) ⊃ (∼Hx ⊃ Cx)]
c. (x){∼Hx ⊃ [(Gx ∨ Sx) ⊃ Cx]}
d. (x)[(Gx • Sx) ⊃ (Cx ⊃ ∼Hx)]
e. (x)[(Gx ∨ Sx) ⊃ (Cx ⊃ ∼Hx)]
9. Large diamonds are costly if they are not flawed.
a. (∃x){∼Fx ⊃ [Lx ⊃ (Dx • Cx)]}
b. (x)[(Lx • Dx) ⊃ (∼Fx ⊃ Cx)]
c. (x)[(Lx ∨Dx) ⊃ (∼Fx ⊃ Cx)]
d. (x)[(Dx • Cx) ⊃ (∼Fx ⊃ Lx)]
e. (x)[(∼Fx ⊃ Cx) ⊃ (Lx • Dx)]
10. Every accountant will be dismissed if any of the books has been fixed.
a. (x)(Ax ⊃Dx) ⊃ (∃x)(Bx • Fx)
b. (x)[Bx • Fx] ⊃ (Ax ⊃ Dx)]
c. (x)[(Bx • Fx) ⊃ (Ax ⊃ Dx)]
d. (∃x)(Bx • Fx) ⊃ (x)(Ax ⊃ Dx)
e. (∃x)[(Bx • Fx) ⊃ (x)(Ax ⊃ Dx)]
11. If all the landscapers are competent, then if none of the roses die, then they will get a bonus.
a. (∃x)(Lx • Cx) ⊃ [(y)(Ry ⊃ ∼Dy) ⊃ Gx]
b. (x){(Lx • Cx) ⊃ [(y)(Ry ⊃ ∼Dy) ⊃ Gx]}
c. (x){(Lx ⊃Cx) ⊃[(y)(Ry ⊃ ∼Dy) ⊃ Gx]}
d. (x)(Lx ⊃ Cx) ⊃[(y)(Ry ⊃ ∼Dy) ⊃ Gx]
e. (x)(Lx ⊃Cx) ⊃(y)[(Ry ⊃ ∼Dy) ⊃ Gy]
12. Melinda read a novel.
a. (∃x)(Nx • Rmx)
b. (∃x)Nx • (∃x)Rmx
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c. (x)(Nx ⊃Rmx)
d. Rmn
e. (∃x)[(Nx • Rx) • Mx]
13. A few athletes excel in every sport.
a. (∃x)[Ax • (Sx ⊃ Ex)]
b. (∃x)[Ax • (y)(Sy ⊃ Exy)]
c. (x)[Ax ⊃(y)(Sy ⊃ Exy)]
d. (∃x)[Ax • (y)Sy ⊃ Exy]
e. (∃x)Ax • (y)(Sy ⊃ Exy)
14. Some musicians can play every tune they hear.
a. (∃x)Mx • [(y)(Ty • Hxy) ⊃ Pxy]
b. (∃x)[Mx • (y)(Ty • Hxy) ⊃ Pxy]
c. (x)Tx ⊃ (∃y)[(My • Hyx) ⊃ Pyx]
d. (x){Mx ⊃[(y)(Ty • Hxy) ⊃ Pxy]}
e. (∃x){Mx • [(y)(Ty • Hxy) ⊃ Pxy]}
15. Every person dislikes someone or other.
a. (x)Px ⊃ (∃y)(Py • Dxy)
b. (∃x)[Px • (y)(Py • Dxy)]
c. (x)[Px ⊃ (∃y)(Py • Dxy)]
d. (∃x)[Px • (∃y)(Py • Dxy)]
e. (x)[Px ⊃ (∃y)Py • Dxy]
16. Susan’s mother is Chris Campbell.
a. (∃x)(Mxs • x = c)
b. (∃x){Mxs • (y)[(Mys ⊃ y = x) ⊃ x = c]}
c. (x)(Mxs ⊃ x = c)
d. (∃x)[Mxs • (y)(Mys ⊃ y = x) • x = c]
e. (x){Mxs ⊃(y)[(Mys ⊃ y = x) • x = c]}
17. There are exactly two cars in the lot.
a. (∃x)(∃y){Cx • Lx • Cy • Ly • x ≠ y • (z)[(Cz • Lz) ⊃ (z = x ∨ z = y)}
b. (∃x)(∃y)(Cx • Lx • Cy • Ly • x ≠ y)
c. (x)(∃y){Cx • Lx • Cy • Ly • x ≠ y • (z)[(Cz • Lz) ⊃ (z = x ∨ z = y)}
d. (∃x)(∃y)(∃z){Cx • Lx • Cy • Ly • x ≠ y • [(Cz • Lz) ⊃ (z = x ∨ z = y)}
e. (x)(y){[Cx • Lx • Cy • Ly • x ≠ y] ⊃ (z)[(Cz • Lz) ⊃ (z = x ∨ z = y)}
18. Evans is the fastest runner on the team.
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a. Re • Te • (∃x)(Rx • Tx • x ≠ e • Fex]
b. Re • Te • (x)[(Rx • Tx • x ≠ e) ⊃ Fex]
c. (x)[(Rx • Tx) ⊃ Fex]
d. Re • Te • (x)[(Rx • Tx) ⊃ Fex]
e. Re • Te • (x)[(Rx • Tx • x = e) ⊃ Fex]
19. Every player except Michael is healthy.
a. Pm • ∼Hm • (∃x)(Px • x ≠ m • Hx)
b. Pm • ∼Hm • (x)(Px • x ≠ m ⊃ Hx)
c. Pm • ∼Hm • (x)[(Px • x ≠ m) ⊃ Hx]
d. (x)(Px • ∼Hx • x ≠ m) ⊃ ∼Hm
e. (∃x)(Px • ∼Hx • x ≠ m) ⊃ (Pm • ∼Hm)
20. Nancy and Raquel will conduct the experiment only if all the young physicists are busy.
a. (Cn • Cr) ⊃ (x)[Bx ⊃ (Yx • Px)]
b. (∃x){Cn • Cr • [(Yx • Px) ⊃ Bx]}
c. (x)[(Yx • Px) ⊃ Bx] ⊃ (Cn • Cr)
d. Nc • Rc • (x)[(Yx • Px) ⊃ Bx]
e. (Cn • Cr) ⊃ (x)[(Yx • Px) ⊃ Bx]
INSTRUCTIONS: Use natural deduction to derive the conclusion in each problem.
21. Use conditional proof or indirect proof as needed:
1. (x)[(Kx ∨Nx) ⊃(Ex • ∼Rx)]
2. (x)[(Kx ∨Sx) ⊃(Rx ∨Hx)] / (x)[Kx ⊃(Ex • Hx)]
22. Use conditional proof or indirect proof as needed:
1. (∃x)(Tx ∨ Gx) ⊃ (x)(Fx ⊃ Mx)
2. (∃x)(Gx • ∼Mx) / (∃x)∼Fx
23. Use conditional proof or indirect proof as needed:
1. ∼(∃x)(Qx • Rx)
2. ∼(∃x)(Px ∨ ∼Qx) / ∼(∃x)(Rx ∨ Px)
24. Use conditional proof or indirect proof as needed:
1. (x)(∃y)[Ax ⊃ (Cy ⊃ Bxy)]
2. (∃x)(y)(Ax • ∼Bxy / ∼(x)Cx
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25. Use conditional proof or indirect proof as needed:
2. (∃x)(Jx • Gc) / a ≠ c
26. Use the finite universe method to prove that the following argument is invalid:
1. (x)(Sx ⊃ Tx)
2. (∃x)∼Tx / (x)∼Sx