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INSTRUCTIONS: Select the correct translation for each statement.
1. Every firefly glows in the dark.
a. (x)(Fx ⊃Gx)
b. (x)(Gx ⊃Fx)
c. (∃x)(Fx ⊃ Gx)
d. (∃x)(Fx • Gx)
e. (∃x)(Hx ⊃ Fx)
2. Alice will cheer if either Casey or Enright scores a touchdown.
a. Ca ⊃ (Sc ∨ Se)
b. (Sc ∨Se) ⊃ Ca
c. (Cs ∨ Es) ⊃ Ac
d. Ac ⊃ (Cs ∨ Es)
e. (∃x)(Cx ∨ Ex) ⊃ (∃y)Ax
3. A mouse is in the closet.
a. (∃x)(Mx ∨ Cx)
b. (x)(Mx ⊃Cx)
c. (∃x)(Mx ⊃ Cx)
d. (x)(Mx • Cx)
e. (∃x)(Mx • Cx)
4. A wallaby is a marsupial.
a. (∃x)(Mx • ∼Wx)
b. (∃x)(Wx • Mx)
c. (∃x)(Wx ⊃ Mx)
d. (x)(Wx ⊃Mx)
e. (x)(Mx ⊃Wx)
5. Not all tennis players are high strung.
a. (x)(Tx ⊃ ∼Hx)
b. (∃x)(Tx ⊃ ∼Hx)
c. (∃x)(Tx • ∼Hx)
d. (x)(Hx ⊃Tx)
e. (x)(Tx • ∼Hx)
6. No liberals are conservatives.
a. (∃x)(Cx • ∼Lx)
b. (∃x)(Lx • ∼Cx)
c. ∼(x)(Lx ⊃ Cx)
d. (∃x)(Lx ⊃ ∼Cx)
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e. (x)(Lx ⊃ ∼Cx)
7. Only frogs and toads inhabit this cave.
a. (x)[Ix ⊃ (Fx ∨ Tx)]
b. (∃x)[(Fx • Tx) ⊃ Ix]
c. (x)[Ix ⊃(Fx • Tx)]
d. (x)[(Fx • Tx) ⊃ Ix]
e. (x)[(Fx ∨ Tx) ⊃ Ix]
8. Every giant sequoia is precious.
a. (x)[(Gx ⊃Px) • (Sx ⊃ Px)]
b. (x)[(Gx • Sx) ⊃ Px]
c. (x)[(Gx ∨Sx) ⊃Px]
d. (x)[Px ⊃ (Gx • Sx)]
e. (∃x)[Gx • Sx) • Px]
9. All the cakes and pies are delicious.
a. (x)[Dx ⊃(Cx ∨Px)]
b. (∃x)[(Px • Cx) • Dx]
c. (x)[(Cx • Px) ⊃ Dx]
d. (x)[(Cx ∨Px) ⊃Dx]
e. (∃x)[(Px • Cx) ⊃ Dx]
10. Miriam will be hired if and only if every manager approves.
a. Hm ≡ (∃x)(Mx • Ax)
b. Hm ≡ (∃x)(Mx ⊃Ax)
c. Hm ≡ (x)(Ax ⊃ Mx)
d. Hm ⊃ (x)(Mx ⊃ Ax)
e. Hm ≡ (x)(Mx ⊃ Ax)
11. If any house burns, then every fireman will respond.
a. (x)(Hx ⊃Bx) ⊃ (∃x)(Fx • Rx)
b. (x)[(Hx • Bx) ⊃ (∃y)(Fy • Ry)]
c. (∃x)(Hx • Bx) ⊃ (x)(Fx ⊃ Rx)
d. (x)(Hx ⊃Bx) ⊃ (∃x)(Fx • Rx)
e. (x)[(Hx • Bx) ⊃ (Fx • Rx)]
12. Angela wrote a poem.
a. (∃x)(Px • Wax)
b. (x)(Wax ⊃Px)
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c. Wap
d. (∃x)(∃y)(Px • Wxy)
e. (∃x)(Px • Ap)
13. A few dogs chase every cat they see.
a. (∃x){Dx ⊃ (y)[(Cy • Sxy) ⊃ Cxy]}
b. (x){Dx ⊃(y)[(Cy • Sxy) ⊃ Cxy]}
c. (∃x)Dx • (y)[(Cy • Sxy) ⊃ Cxy]
d. (∃x)Dx ⊃ (y)[(Cy • Sxy) ⊃ Cxy
e. (∃x){Dx • (y)[(Cy • Sxy) ⊃ Cxy]}
14. If there are any guards, then if none of the prisoners escape, then they will be rewarded.
a. (x){[Gx ⊃(y)(Py ⊃ ∼Ey)] ⊃ Rx}
b. (x){Gx ⊃ [(∃y)(Py • ∼Ey) ⊃ Rx]}
c. (∃x)Gx ⊃ [(y)(Py ⊃ ∼Ey) ⊃ Rx]
d. (x){Gx ⊃ [(y)(Py ⊃ ∼Ey) ⊃ Rx]}
e. (∃x){Gx • [(y)(Py ⊃ ∼Ey) ⊃ Rx]}
15. Every person trusts someone or other.
a. (∃x)Px • (∃y)(Py • Txy)
b. (x)[Px ⊃ (∃y)(Py • Txy)]
c. (x)[Px ⊃ (y)(Py ⊃ Txy)]
d. (∃x)[Px • (∃y)(Py • Txy)]
e. (x)Px ⊃ (∃y)(Py • Txy)
16. If every witness tells the truth, then none of the guilty defendants will be acquitted.
a. (x){(Wx ⊃ Tx) ⊃ (x)[(Gx • Dx) ⊃ ∼Ax]}
b. (x)(Wx ⊃ Tx) ⊃ (∃x)[Gx • (Dx ⊃ ∼Ax)]
c. (x)(Wx ⊃ Tx) ⊃ (x)[(Gx • Dx) ⊃ ∼Ax]
d. (x){(Wx ⊃ Tx) ⊃ [(Gx • Dx) ⊃ ∼Ax]}
e. (∃x)(Wx • Tx) ⊃ (x)[(Gx • Dx) ⊃ ∼Ax]
17. The capital of Arkansas is not Saint Louis.
a. (∃x)(Cxa • x ≠ s)
b. (x)(Cxa ⊃ x ≠ s)
c. (∃x)[Cxa • (y)(Cya ⊃ x ≠ s)]
d. (∃x)[Cxa • (y)(Cya ⊃ y = x) • x ≠ s]
e. (∃x)Cxa • ∼Csa
18. The only victim who survived is Oliver.
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a. Vo • So • (x)[(Vx • Sx) ⊃ x = o]
b. (Vo • So) ⊃ (x)[(Vx • Sx) ⊃ x = o]
c. (x)[(Vx • Sx) ⊃ x = o]
d. (∃x)[(Vx • Sx) • x = o]
e. (∃x)(Vx • Sx) ⊃ (x)(x = o)
19. Every city except Edenville was flooded.
a. (x)[(Cx • x ≠ e) ⊃ Fx]
b. Ce • ∼Fe • (x)[(Cx • x ≠ e) ⊃ Fx]
c. (Ce • ∼Fe) ⊃(x)[(Cx • x ≠ e) ⊃ Fx]
d. Ce • ∼Fe • (∃x)[(Cx • x ≠ e) • Fx]
e. Ce • ∼Fe • (x)[(Cx • Fx) ⊃ x ≠ e]
20. Rollins is the shortest player on the team.
a. Pr • (∃x)[(Px • x ≠ r) • Srx]
b. Pr • (x)(Srx ⊃ x ≠ r)
c. Pr • (x)(Px ⊃ Srx)
d. Pr ⊃ (x)[(Px • x ≠ r) ⊃ Srx]
e. Pr • (x)[(Px • x ≠ r) ⊃ Srx]
INSTRUCTIONS: Use natural deduction to derive the conclusion in each problem.
21. Use conditional proof or indirect proof as needed:
1. (x)[Hx ⊃ (Rx • Tx)]
2. (x)[(Rx ∨Hx) ⊃(Gx • Fx)] / (x)(Hx ⊃Fx)
22. Use conditional proof or indirect proof as needed:
1. (x)[Kx ⊃(Bx • Cx)]
2. (∃x)∼Cx / (∃x)∼Kx
23. Use conditional proof or indirect proof as needed:
1. (∃x)(Sx ∨ Qx) ⊃ (x)[Qx ⊃ (Nx ∨ Dx)]
2. (∃x)(Qx • ∼Dx) / (∃x)Nx
24. Use conditional proof or indirect proof as needed:
1. (x)(∃y)(Ax ∨ Nxy)
2. (∃x)(y)(Ax ⊃ Nxy) / (∃x)(∃y)Nxy
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25. Use conditional proof or indirect proof as needed:
1. (x)(∃y)(Gx ⊃ Hay)
2. (∃x)(y)(Gx • ∼Hey) / a ≠ e
26. Use the finite universe method to prove that the following argument is invalid:
1. (x)(Ax ∨Bx)
2. (∃x)∼Ax / (x)Bx