Deck 8: Predicate Logic

Question 1

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)]

Answer 1

(x)(Gx ⊃Hx) ⊃(x)[(Rx ⊃ (Cx • Tx)]

Question 2

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]

Answer 2

Dr • Sr • (x)[(Dx • Sx • x ≠ r) ⊃ Brx]

Question 3

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]

Answer 3

(x)(y)[(Wx • Wy) ⊃ x = y]

Question 4

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

Question 5

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)

Question 6

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)

Question 7

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]}

Question 8

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)

Question 9

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

Question 10

Whoever rides horses is adventurous.

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]

Question 11

A few scholarships were awarded.

A) (∃x)(Sx ⊃ Ax)
B) (∃x)(Sx • Ax)
C) (x)(Sx ⊃Ax)
D) Sy • Ay
E) (x)(Sx • Ax)

Question 12

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]

Question 13

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)

Question 14

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)

Question 15

A freshman is not a sophomore.

A) Fy ⊃ ∼Sy
B) (∃x)(Fx • Sx)
C) (∃x)(Fx • ∼Sx)
D) Fx • ∼Sx
E) (x)(Fx ⊃ ∼Sx)

Question 16

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)

Question 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]

Question 18

The only daughter Robert has is Esther.

A) (x)(Dxr ⊃ e = e)
B) Der • (x)(Dxr ⊃ x = e)
C) Der
D) Der • (x)(x = e ⊃ Der)
E) (∃x)(Dxr • x = e)

Question 19

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)].

Question 20

A taxi is waiting.

A) (x)(Tx ⊃Wx)
B) (∃x)(Tx ⊃ Wx)
C) (∃x)(Tx • Wx)
D) (x)(Tx • Wx)
E) Tx • Wx

Question 21

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]

Question 22

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)

Question 23

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]

Question 24

The only victim who survived is Oliver.

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)

Question 25

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]

Question 26

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)

Question 27

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)

Question 28

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]

Question 29

No liberals are conservatives.

A) (∃x)(Cx • ∼Lx)
B) (∃x)(Lx • ∼Cx)
C) ∼(x)(Lx ⊃ Cx)
D) (∃x)(Lx ⊃ ∼Cx)
E) (x)(Lx ⊃ ∼Cx)

Question 30

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

Question 31

Angela wrote a poem.

A) (∃x)(Px • Wax)
B) (x)(Wax ⊃Px)
C) Wap
D) (∃x)(∃y)(Px • Wxy)
E) (∃x)(Px • Ap)

Question 32

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]}

Question 33

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]}

Question 34

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)

Question 35

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)]

Question 36

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)

Question 37

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)

Question 38

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

Question 39

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]

Question 40

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]

Question 41

Melinda read a novel.

A) (∃x)(Nx • Rmx)
B) (∃x)Nx • (∃x)Rmx
C) (x)(Nx ⊃Rmx)
D) Rmn
E) (∃x)[(Nx • Rx) • Mx]

Question 42

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]

Question 43

Evans is the fastest runner on the team.

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]

Question 44

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)}

Question 45

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

Question 46

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)]

Question 47

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)]

Question 48

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)

Question 49

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]}

Question 50

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]

Question 51

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)

Question 52

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]

Question 53

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

Question 54

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]

Question 55

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)

Question 56

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)]

Question 57

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)]
E) (x)[(Bx ∨Lx) ⊃(Mx • Cx)]

Question 58

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]}

Question 59

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]

Question 60

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)]