Quantifiers questions

Question: Every person trusts someone or other.

Answer Options:
a. (x)(Px ⊃ (y)(Ty ⊃ 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)]

Answer: b. (x)(Px ⊃ (∃y)(Py ⊃ Txy)]

Question: Alice will cheer if either Casey or Enright scores a touchdown.

Answer Options:
a. Ca ⊃ (Sc ∨ Se)
b. (Sc ∨ Se) ⊃ Ca
c. (Cv ⊃ Se) ⊃ Ac
d. Ac ⊃ (Cs ∨ Es)
e. (∃x)(Cv ∨ Ex) ⊃ (∃y)Ax

Answer: b. (Sc ∨ Se) ⊃ Ca

Question: If all the plumbers are skilled, then if none of the faucets leak, then they will be commended.

Answer Options:
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) • ∼Ly) ⊃ Cx)]

Answer: d. (x)(Px • Sx) ⊃ [(y)(Fy ⊃ ∼Ly) ⊃ Cx)]

Question: The father of Angelo is an Italian.

Answer Options:
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)

Answer: c. (∃x)((Fxa • (y)(Fya ⊃ y = x) ⊃ Ix)]

Question: If any house burns, then every fireman will respond.

Answer Options:
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) ⊃ (y)(Fx • Rx)]
e. (x)((Hx ⊃ Bx) ⊃ (y)(Fx • Rx)]

Answer: c. (x)(Hx ⊃ Bx) ⊃ (∃x)(Fx ⊃ Rx)

Question: The biggest dog in the show is Rover.

Answer Options:
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]

Answer: d. Dr • Sr • (x)[(Dx • Sx ⊃ x ≠ r)] ⊃ Brx]

Question: Not every applicant is eligible.

Answer Options:
a. (∃x)(Ax ⊃ Ex)
b. (∃x)(Ax • (∃x)∼Ex
c. (x)(Ax ⊃ ∼Ex)
d. (x)(Ax ⊃ Ex)
e. (x)(Ex ⊃ Ax)

Answer: d. (x)(Ax ⊃ Ex)

Question: Every journalist knows how to write.

Answer Options:
a. Jx ⊃ Kx
b. (∃x)(Jx ⊃ Kx)
c. (∃x)Jx ⊃ (∃x)Kx
d. (x)(Jx ⊃ Kx)
e. (∃y)(Jy • Ky)

Answer: d. (x)(Jx ⊃ Kx)

Question: Every giant sequoia is precious.

Answer Options:
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]

Answer: b. (x)[(Gx ∨ Sx) ⊃ Px]

Question: Whoever rides horses is adventurous.

Answer Options:
a. (x)[Px ⊃ (∃y)(Hy ⊃ Rxy) ⊃ Ax]
b. (x)[Px ⊃ (∃y)(Hy ⊃ Rxy) ⊃ Ax]
c. (x)[Px ⊃ (y)(Hy ⊃ Rxy) ⊃ Ax]
d. (∃x)(Px • Rxh) ⊃ (∃y)Ay
e. (x)[Px ⊃ (∃y)(Hy ⊃ Rxy) ⊃ Ax]

Answer: b. (x)[Px ⊃ (∃y)(Hy ⊃ Rxy) ⊃ Ax]

Question: The capital of Arkansas is not Saint Louis.

Answer Options:
a. (∃x)(Cxa • x ≠ s)
b. (x)(Cxa ≠ x ≠ s)]
c. (∃x)(Cxa ⊃ (y)(Cya ≠ y = x • x ≠ s)]
d. (∃x)(Cxa ⊃ (Cya • ≠Csa)]

Answer: d. (∃x)(Cxa ⊃ (Cya • ≠Csa)]

Question: If Andy and Carol pass the test, then Eve will be delighted.

Answer Options:
a. (Pa ∨ Pc) ⊃ Ed
b. (x)Px ⊃ (∃y)Dy
c. (Ap • Cp) ⊃ Ed
d. Pac ⊃ De
e. (Pa • Pc) ⊃ De

Answer: e. (Pa • Pc) ⊃ De

Question: Only frogs and toads inhabit this cave.

Answer Options:
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]

Answer: a. (x)[Ix ⊃ (Fx ∨ Tx)]

Question: Every student except Christopher passed the course.

Answer Options:
a. (∃x)[Sx • ∼Px ⊃ x = c]
b. (Sc • ∼Pc) ⊃ (x)[(Sx • Px) ⊃ x ≠ c)]
c. (x)[(Sx • Px) ⊃ (Sx • c) ⊃ ∼Px)]
d. (x)[(Sx • Px) ⊃ x ≠ c]
e. Sc • ∼Pc ⊃ (x)[(Sx • x ≠ c) ⊃ Px]

Answer: e. Sc • ∼Pc ⊃ (x)[(Sx • x ≠ c) ⊃ Px]

Question: If every witness tells the truth, then none of the guilty defendants will be acquitted.

Answer Options:
a. (x)((Wx ⊃ Tx) ⊃ (y)((Gx ⊃ Dx) ⊃ ∼Ax)]
b. (x)((Wx ⊃ Tx) ⊃ (Gx ⊃ (Dx ⊃ ∼Ax)]
c. (x)((Wx ⊃ Tx) ⊃ (y)((Gx ⊃ Dx) ⊃ ∼Ax)]
d. (x)((Wx ⊃ Tx) ⊃ (y)((Gx ⊃ Dx) ⊃ Ax)]
e. (∃x)(Wx • Tx) ⊃ (y)((Gx ⊃ Dx) ⊃ ∼Ax)]

Answer: c. (x)((Wx ⊃ Tx) ⊃ (y)((Gx ⊃ Dx) ⊃ ∼Ax)]