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Chapter 08 Test C
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INSTRUCTIONS: Select the correct translation for each statement.
1. Megan is a biologist only if William is an astronomer.
a. Bm ⊃ Aw
b. Mb ⊃ Wa
c. Aw ⊃ Bm
d. (∃x)Bx ⊃ (∃x)Ax
e. (x)(Mx ⊃Wx)
2. If Andy and Carol pass the test, then Eve will be delighted.
a. (Pa ∨Pc) ⊃ Ed
b. (x)Px ⊃ (∃y)Dy
c. (Ap • Cp) ⊃ Ed
d. Pac ⊃ De
e. (Pa • Pc) ⊃ De
3. Every journalist knows how to write.
a. Jx ⊃ Kx
b. (∃x)(Jx ⊃ Kx)
c. (∃x)Jx ⊃ (∃x)Kx
d. (x)(Jx ⊃Kx)
e. (∃y)(Jy • Ky)
4. A few scholarships were awarded.
a. (∃x)(Sx ⊃ Ax)
b. (∃x)(Sx • Ax)
c. (x)(Sx ⊃Ax)
d. Sy • Ay
e. (x)(Sx • Ax)
5. A freshman is not a sophomore.
a. Fy ⊃ ∼Sy
b. (∃x)(Fx • Sx)
c. (∃x)(Fx • ∼Sx)
d. Fx • ∼Sx
e. (x)(Fx ⊃ ∼Sx)
6. A taxi is waiting.
a. (x)(Tx ⊃Wx)
b. (∃x)(Tx ⊃ Wx)
c. (∃x)(Tx • Wx)
d. (x)(Tx • Wx)
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e. Tx • Wx
7. Not every applicant is eligible.
a. (∃x)(Ax • Ex)
b. (∃x)Ax • (∃x)∼Ex
c. (x)(Ax ⊃ ∼Ex)
d. (∃x)(Ax • ∼Ex)
e. (x)(Ex ⊃Ax)
8. Ivan will be sad if and only if any child is injured.
a. Si ≡ (∃x)(Cx • Ix)
b. (x)[(Cx • Ix) ≡ Ix]
c. (∃x)(Cx • Ix) ⊃ Si
d. Si ≡ (x)(Cx • Ix)
e. Si ≡ (x)(Cx ⊃ Ix)
9. Elms and maples are deciduous trees.
a. (x)[(Ex ∨Mx) ⊃ (Dx ⊃ Tx)]
b. (∃x)[(Ex ∨ Mx) ⊃ (Tx • Dx)]
c. (x)[(Ex • Mx) ⊃ (Tx • Dx)]
d. (x)[(Ex ∨Mx) ⊃(Tx • Dx)]
e. (∃x)[(Ex ∨ Mx) ⊃ (Tx • Dx)]
10. The guests will be happy only if every room is clean and tidy.
a. (∃x)(Gx • Hx) ⊃ (∃x)[Rx ⊃ (Cx • Tx)]
b. (x)[Rx ⊃(Cx • Tx)] ⊃ (∃x)(Gx • Hx)
c. (∃x)(Gx • Hx) ⊃ (x)[(Rx ⊃ (Cx • Tx)]
d. (x){(Gx ⊃ Hx) ⊃[(Rx ⊃(Cx • Tx)]}
e. (x)(Gx ⊃Hx) ⊃(x)[(Rx ⊃ (Cx • Tx)]
11. If Nancy marries Ralph, then everyone in the family will be happy.
a. (Mn • Rm) ⊃ (x)(Fx ⊃ Hx)
b. (∃x)(Fx • Hx) ⊃ Mnr
c. Mnr ⊃(x)(Fx ⊃ Hx)
d. (Nm • Rm) ⊃ (x)(Fx ⊃ Hx)
e. Mnr ⊃ (∃x)(Fx ⊃ Hx)
12. Whoever rides horses is adventurous.
a. (x){Px • (∃y)[(Hy • Rxy) ⊃ Ax]}
b. (x){[Px • (∃y)[(Hy • Rxy)] ⊃ Ax}
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c. (x){[Px • (y)[(Hy • Rxy)] ⊃ Ax}
d. (∃x)(Px • Rxh) ⊃ (∃y)Ay
e. (x)Px ⊃ (∃y)[(Hy • Rxy) ⊃ Ax]
13. If all the plumbers are skilled, then if none of the faucets leak, then they will be commended.
a. (x){(Px • Sx) ⊃ [(y)(Fy ⊃ ∼Ly) ⊃ Cx]}
b. (∃x)(Px • Sx) ⊃ [(y)(Fy ⊃ ∼Ly) ⊃ Cy]
c. (x)(Px • Sx) ⊃ [(y)(Fy ⊃ ∼Ly) ⊃ Cx]
d. (x){(Px ⊃Sx) ⊃[(y)(Fy ⊃ ∼Ly) ⊃ Cx]}
e. (x){(Px • Sx) ⊃ [(∃y)(Fy • ∼Ly) ⊃ Cx]}
14. Some children lose every toy they own.
a. (∃x){Cx • (y)[(Ty • Oxy) ⊃ Lxy]}
b. (∃x){[Cx • (∃y)(Ty • Oxy)] ⊃ Lxy}
c. (x){[Cx • (y)(Ty • Oxy)] ⊃ Lxy}
d. (x){[Cx ⊃(y)(Ty • Oxy)] ⊃ Lxy}
e. (∃x)[Cx • (y)(Ty • Oxy)] ⊃ (y)Lxy
15. Everyone fears someone (or other).
a. (x)(y)[(Px • Py) ⊃ Fxy]
b. (x)(∃y)[(Px • Py) ⊃ Fxy]
c. (∃x)(y)[(Px • Py) • Fxy]
d. (x)(∃y)[(Px • Py) • Fxy]
e. (∃x)(∃y)[(Px • Py) • Fxy]
16. The father of Angelo is an Italian.
a. (x){[Fxa • (y)(Fya ⊃ y = x)] ⊃ Ix}
b. (∃x)(Fxa • Ix)
c. (∃x)[Fxa • (y)(Fya ⊃ y = x) • Ix]
d. (∃x)[Fxa • (y)(Fya ⊃ y = x)] ⊃ (∃x)Ix
e. (x)(Fxa ⊃Ix)
17. Every student except Christopher passed the course.
a. (∃x)[Sx • ∼Px • x = c]
b. (Sc • ∼Pc) ⊃(x)[(Sx • Px) ⊃ x ≠ c]
c. Sc • ∼Pc • (x)[(Sx • x = c) ⊃ ∼Px]
d. (x)[(Sx • ∼Px) ⊃ x = c]
e. Sc • ∼Pc • (x)[(Sx • x ≠ c) ⊃ Px]
18. The only daughter Robert has is Esther.
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a. (x)(Dxr ⊃ e = e)
b. Der • (x)(Dxr ⊃ x = e)
c. Der
d. Der • (x)(x = e ⊃ Der)
e. (∃x)(Dxr • x = e)
19. There is at most one winner.
a. (∃x)(∃y)(Wx • Wy)
b. (x)(y)[(Wx • Wy) ≡ x = y]
c. (x)(y)[(Wx • Wy) ⊃ x = y]
d. (∃x)(∃y)[(Wx • Wy) • x = y]
e. (x)(∃y)[(Wx • Wy) • e = y]
20. The biggest dog in the show is Rover.
a. Dr • Sr • (x)[(Dx • Sx • Brx) ⊃ x ≠ r]
b. Dr • Sr • (x)[(Dx • Sx) ⊃ Brx]
c. (x)[(Dx • Sx • x ≠ r) ⊃ Brx]
d. Dr • Sr • (x)[(Dx • Sx • x ≠ r) ⊃ Brx]
e. (x)[(Dx • Sx) ⊃ Brx]
INSTRUCTIONS: Use natural deduction to derive the conclusion in each problem.
21. Use conditional proof or indirect proof as needed:
1. (x)[(Sx ∨Nx) ⊃(Fx ∨Hx)]
2. (∃x)(Nx • ∼Hx) / (∃x)Fx
22. Use conditional proof or indirect proof as needed:
1. (x)[Rx ⊃(Tx • ∼Ex)]
2. (x)[(Qx • Rx) ⊃ Ex] / (x)(Rx ⊃ ∼Qx)
23. Use conditional proof or indirect proof as needed:
1. (x)Gx ∨(x)(Bx ⊃ Gx)
2. ∼(x)(Gx ∨ Kx) / ∼(x)(Bx ∨ Kx)
24. Use conditional proof or indirect proof as needed:
1. (x)(∃y)(Tx • ∼Nxy)
2. (∃y)(x)[Tx ⊃ (Nxy ∨ Rxy)] / (∃x)(∃y)Rxy
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25. Use conditional proof or indirect proof as needed:
2. (x)[Ex ⊃(Dx • x = e)] / (∃x)(Ex ⊃ a = e)
26. Use the finite universe method to prove that the following argument is invalid:
1. (x)Ax ⊃ (∃x)Bx
2. (∃x)Ax / (∃x)Bx