Predicate Logic questions and answers

Question: Evans is the fastest runner on the team.

Answer Options:
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]

Answer: b. Re • Te ⊃ (x)[(Rx • Tx • x ≠ e) ⊃ Fex]

Question: Every accountant will be dismissed if any of the books has been fixed.

Answer Options:
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) ⊃ (Ax ⊃ Dx)]

Answer: d. (∃x)(Bx • Fx) ⊃ (x)(Ax ⊃ Dx)

Question: If all the landscapers are competent, then if none of the roses die, then they will get a bonus.

Answer Options:
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]]

Answer: c. (x)[(Lx ⊃ Cx) ⊃ (y)[(Ry ⊃ ∼Dy) ⊃ Gx]]

Question: A raccoon is not a mongoose.

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

Answer: a. (x)(Rx ⊃ ∼Mx)

Question: Susan’s mother is Chris Campbell.

Answer Options:
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]

Answer: d. (∃x)[Mxs ⊃ (y)(Mys ⊃ y = x) • x = c]

Question: Nancy and Raquel will conduct the experiment only if all the young physicists are busy.

Answer Options:
a. (Cn • Cr) ⊃ (x)(Bx ⊃ (Yx ⊃ Px)]
b. (∃x)(Cn • Cr ⊃ [(Yx • Px) ⊃ Bx)]
c. (x)[(Px • Bx) ⊃ (Cn • Cr)]
d. Nc • Rc ⊃ [(Yx • Px) ⊃ Bx)]
e. (Cn • Cr) ⊃ [(Yx • Px) ⊃ Bx)]

Answer: e. (Cn • Cr) ⊃ [(Yx • Px) ⊃ Bx)]

Question: There is a lamp in the bedroom.

Answer Options:
a. (∃x)(Lx ⊃ Bx)
b. (x)(Lx ⊃ Bx)
c. (∃x)(Lx • Bx)
d. (∃x)Lx • (∃x)Bx
e. (∃x)Lx ⊃ (∃x)Bx

Answer: d. (∃x)Lx • (∃x)Bx

Question: None but the experienced drivers are cautious and safe.

Answer Options:
a. (x)[Dx ⊃ (Ex ⊃ (Cx • Sx))]
b. (x)[(Ex • Dx) ⊃ (Cx • Sx)]
c. (x)[(Cx • Sx) ⊃ Ex)]
d. (x)[Dx ⊃ (Cx ⊃ Ex)]
e. (∃x)[(Dx • Cx) ⊃ (Ex • Sx)]

Answer: d. (x)[Dx ⊃ (Cx ⊃ Ex)]

Question: Every book in the library is misplaced or checked out.

Answer Options:
a. (x)[(Bx • Lx) ⊃ (Mx ∨ Cx)]
b. (x)[(Bx • Mx) ⊃ Cx)]
c. (x)[(Bx ⊃ Lx) ⊃ Mx]
d. (x)[(Bx • Lx) ⊃ (Mx • Cx)]
e. (x)[(Bx ∨ Lx) ⊃ (Mx • Cx)]

Answer: a. (x)[(Bx • Lx) ⊃ (Mx ∨ Cx)]

Question: There are exactly two cars in the lot.

Answer Options:
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 • (z)[(Cz • Lz) ⊃ (z = x ∨ z = y)]
e. (x)(∃y)[(Cx • Lx • Cy • Ly • x ≠ y] ⊃ (z)[(Cz • Lz) ⊃ (z = x ∨ z = y)]

Answer: a. (∃x)(∃y)[Cx • Lx • Cy • Ly • x ≠ y • (∃z)[(Cz • Lz) ⊃ (z = x ∨ z = y)]

Question: Large diamonds are costly if they are not flawed.

Answer Options:
a. (x)[Lx ⊃ Fx ⊃ (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)]

Answer: b. (x)[(Lx • Dx) ⊃ (∼Fx ⊃ Cx)]

Question: A few athletes excel in every sport.

Answer Options:
a. (∃x)[Ax • ($x ⊃ Exy)]
b. (∃x)[Ax • (y)(Sy ⊃ Exy)]
c. (x)(Ax ⊃ (y)(Sy ⊃ Exy)]
d. (∃x)Ax ⊃ (y)(Sy ⊃ Exy)]
e. (∃x)Ax • (y)(Sy ⊃ Exy)

Answer: b. (∃x)[Ax • (y)(Sy ⊃ Exy)]

Question: Goats and sheep are contented only if they are not hungry.

Answer Options:
a. (x)[(Gx • Sx) ⊃ (Hx ⊃ Cx)]
b. (x)[(Gx ∨ Sx) ⊃ ∼Hx ⊃ Cx)]
c. (∃x)[(Hx ⊃ ∼Cx) ⊃ (Gx ⊃ Sx)]
d. (x)[∼Hx ⊃ (Gx ⊃ Sx)]
e. (x)[(Gx ∨ Sx) ⊃ (Hx ⊃ ∼Cx)]

Answer: e. (x)[(Gx ∨ Sx) ⊃ (Hx ⊃ ∼Cx)]

Question: Some musicians can play every tune they hear.

Answer Options:
a. (∃x)Mx ⊃ [(y)(Ty • Hxy) ⊃ Pxy]
b. (∃x)Mx ⊃ (y)(Ty ⊃ Hxy) ⊃ Pxy]
c. (x)Tx ⊃ (y)(Ty ⊃ Mx) ⊃ Pxy]
d. (x)(Mx ⊃ (Ty • Hxy ⊃ Pxy)]
e. (∃x)(Mx ⊃ (y)(Ty ⊃ Pxy)]

Answer: e. (∃x)(Mx ⊃ (y)(Ty ⊃ Pxy)]

Question: Not every model is emaciated.

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

Answer: c. (∃x)(Mx • ∼Ex)

Question: Every player except Michael is healthy.

Answer Options:
a. Pm • ∼Hm ⊃ (x)(Px • x ≠ m ⊃ Hx)
b. Pm • ∼Hm ⊃ (x)(Px • x ≠ m ⊃ Hx)
c. (∃x)(Px • ∼Hx • x ≠ m) ⊃ (Pm ⊃ ∼Hm)
d. (x)(Px • ∼Hx • x ≠ m) ⊃ (Pm • ∼Hm)
e. (∃x)(Px • ∼Hx • x ≠ m) ⊃ (Pm • ∼Hm)

Answer: c. (∃x)(Px • ∼Hx • x ≠ m) ⊃ (Pm ⊃ ∼Hm)

Question: A small bird landed on the roof.

Answer Options:
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]

Answer: e. (∃x)[(Sx • Bx) ⊃ Lx]

Question: If either Ann or Charlie wins the lottery, then George will celebrate.

Answer Options:
a. (x)[Ax ∨ Cx) ⊃ Gx]
b. (Wa ∨ Wc) ⊃ Gg
c. (Wa ∨ Wc) ⊃ Cg
d. (∃x)[(Ax ∨ Cx) • Gx]
e. (∃x)[(Wa ∨ Wc) ⊃ Cg]

Answer: b. (Wa ∨ Wc) ⊃ Gg

Question: Every person dislikes someone or other.

Answer Options:
a. (∃x)[Px ⊃ (y)(Py ⊃ Dxy)]
b. (∃x)[Px ⊃ (y)(Py ⊃ Dxy)]
c. (x)[Px ⊃ (∃y)(Py ⊃ Dxy)]
d. (∃x)[Px ⊃ (Py ⊃ Dxy)]
e. (x)[Px ⊃ (∃y)Py • Dxy)]

Answer: c. (x)[Px ⊃ (∃y)(Py ⊃ Dxy)]