Deck 5: Full First-Order Logic

Question 1

select the best translation into predicate logic, using the following translation key:
h: Hume
l: Locke
Px: x is a philosopher
Rx: x is a rationalist
Ixy: x influenced y
Sxy: x is more skeptical than y
-Hume is more skeptical than Locke.

A) Slh
B) Shl
C) hSl
D) lSh
E) hlS

Answer 1

B

Question 2

select the best translation into predicate logic, using the following translation key:
h: Hume
l: Locke
Px: x is a philosopher
Rx: x is a rationalist
Ixy: x influenced y
Sxy: x is more skeptical than y
-Locke influenced Hume, but is not more skeptical than him.

A) Ilh • ∼Slh
B) Ihl • ~Shl
C) Ilh • ~Shl
D) (I • ~S)lh
E) Ilh $\lor$ Shl

Answer 2

A

Question 3

select the best translation into predicate logic, using the following translation key:
h: Hume
l: Locke
Px: x is a philosopher
Rx: x is a rationalist
Ixy: x influenced y
Sxy: x is more skeptical than y
-No philosopher is more skeptical than Hume.

A) (∀x)(Px ⊃ ∼Sxh)
B) ~(∃x)(Px • Shx)
C) (∀x)~(Px ⊃ Sxh)
D) (∀x)(~Sxh ⊃ Px)
E) (∀x)(Shx ⊃ Px)

Answer 3

A

Question 4

select the best translation into predicate logic, using the following translation key:
h: Hume
l: Locke
Px: x is a philosopher
Rx: x is a rationalist
Ixy: x influenced y
Sxy: x is more skeptical than y
-Hume influenced some philosophers more skeptical than Locke.

A) (∀x)[Px ⊃ (Sxl • Ihx)]
B) (∃x)[(Px • Ihx) ⊃ Sxl]
C) (∀x)[(Sxl • Ihx) ⊃ Px]
D) (∃x)[(Px • Slx) ⊃ ~Ihx]
E) (∃x)[(Px • Sxl) • Ihx]

Question 5

select the best translation into predicate logic, using the following translation key:
h: Hume
l: Locke
Px: x is a philosopher
Rx: x is a rationalist
Ixy: x influenced y
Sxy: x is more skeptical than y
-All philosophers are more skeptical than some rationalists.

A) (∀x)[Px ⊃ (∀y)(Ry ⊃ Sxy)]
B) (∀x)[Px ⊃ (∀y)(Sxy ⊃ Ry)]
C) (∀x)[Px • (∃y)(Ry • Sxy)]
D) (∀x)[Px ⊃ (∃y)(Ry • Sxy)]
E) (∃x)[Px • (∃y)(Ry • Sxy)]

Question 6

select the best translation into predicate logic, using the following translation key:
h: Hume
l: Locke
Px: x is a philosopher
Rx: x is a rationalist
Ixy: x influenced y
Sxy: x is more skeptical than y
-Hume is more skeptical than all rationalist philosophers.

A) (∀x)[Sxh ⊃ (Px • Rx)]
B) (∀x)[(Px • Rx) ⊃ Shx]
C) (∀x)[(Px • Rx) ⊃ Sxh]
D) (∀x)[Shx ⊃ (Px • Rx)]
E) (∀x)[(Px • Rx) ≡ Shx]

Question 7

select the best translation into predicate logic, using the following translation key:
h: Hume
l: Locke
Px: x is a philosopher
Rx: x is a rationalist
Ixy: x influenced y
Sxy: x is more skeptical than y
-Hume is not more skeptical than some philosophers.

A) ~(∃x)(Px • Sxh)
B) (∃x)(Px • ∼Shx)
C) ~(∀x)(Px • Shx)
D) (∃x)(Px ⊃ ~Shx)
E) (∃x)(~Shx ⊃ Px)

Question 8

select the best translation into predicate logic, using the following translation key:
h: Hume
l: Locke
Px: x is a philosopher
Rx: x is a rationalist
Ixy: x influenced y
Sxy: x is more skeptical than y
-Some philosophers more skeptical than Hume influenced all philosophers more skeptical
Than Locke.

A) (∃x){(Px • Sxh) • (∀y)[(Py • Syl) ⊃ Ixy]}
B) (∃x){(Px • Sxh) • (∀y)[(Py • Sly) ⊃ Ixy]}
C) (∃x){(Px • Shx) • (∀y)[(Py • Sly) ⊃ Ixy]}
D) (∃x){(Px • Shx) • (∀y)[(Py • Sly) ⊃ Iyx]}
E) (∃x){(Px • Sxh) • (∀y)[(Py • Sly) ⊃ Iyx]}

Question 9

select the best translation into predicate logic, using the following translation key:
h: Hume
l: Locke
Px: x is a philosopher
Rx: x is a rationalist
Ixy: x influenced y
Sxy: x is more skeptical than y
-Some rationalists influenced some philosophers who were more skeptical than them.

A) (∀x){Px ⊃ (∃y)[(Ry • Syx) • Ixy]}
B) (∃x){Px • (∃y)[(Ry • Sxy) • Ixy]}
C) (∃x){Rx • (∃y)[(Py • Syx) • Ixy]}
D) (∃x){Rx • (∃y)[(Py • Sxy) • Iyx]}
E) (∃x){Rx • (∀y)[(Py • Syx) ⊃ Ixy]}

Question 10

select the best translation into predicate logic, using the following translation key:
h: Hume
l: Locke
Px: x is a philosopher
Rx: x is a rationalist
Ixy: x influenced y
Sxy: x is more skeptical than y
-If some rationalist is more skeptical than Locke, then no philosopher influenced Hume.

A) (∃x)(Rx • Sxl) ⊃ (∀x)(Px ⊃ ∼Ixh)
B) (∃x)[(Rx • Sxl) ⊃ (∀y)(Py ⊃ ~Iyh)]
C) (∃x)[(Rx • Sxl) ⊃ ~(∃y)(Py ⊃ ~Iyh)]
D) (∃x)(Rx • Sxl) ⊃ ~(∃x)(Px ⊃ ~Ixh)
E) (∃x)[(Rx • Sxl) ⊃ (Px ⊃ ~Ixh)]

Question 11

select the best translation into predicate logic, using the following translation key:
Cx: x is a cheetah
Lx: x is a lion
Tx: x is a tiger
Fxy: x is faster than y
Lxy: x is larger than y
-Some cheetahs are faster than all lions.

A) (∀x)[Lx ⊃ (∃y)(Cy • Fxy)]
B) (∀x)[Lx ⊃ ~(∀y)(Cy • Fyx)]
C) (∃x)(Cx ⊃ (∀y)(Ly ⊃ Fxy)]
D) (∃x)[Cx • (∀y)(Ly ⊃ Fxy)]
E) (∃x)[Cx • (∀y)(Ly • Fxy)]

Question 12

select the best translation into predicate logic, using the following translation key:
Cx: x is a cheetah
Lx: x is a lion
Tx: x is a tiger
Fxy: x is faster than y
Lxy: x is larger than y
-No tiger is faster than all cheetahs.

A) ∼(∃x)[Tx • (∀y)(Cy ⊃ Fxy)]
B) (∀x)[Tx ⊃ ~(∀y)(Cy • Fxy)]
C) (∀x)[Tx ⊃ ~(∃y)(Cy ⊃ Fxy)]
D) ~(∃x)[Cx • (∃y)(Ty . Fyx)]
E) ~(∃x)[Tx • (∀y)(Fyx ⊃ Cy)]

Question 13

select the best translation into predicate logic, using the following translation key:
Cx: x is a cheetah
Lx: x is a lion
Tx: x is a tiger
Fxy: x is faster than y
Lxy: x is larger than y
-No lion is faster than some tigers.

A) ~(∃x)[Lx • (∀y)(Ty ⊃ Fyx)]
B) (∀x)[Lx ⊃ (∃y)(Ty • ∼Fxy)]
C) ~(∃x)[Lx • (∃y)(Ty • Fxy)]
D) ~(∃x)(∃y)[(Lx • Ty) • Fxy]
E) (∀x)[Lx ⊃ (∀y)(Ty ⊃ ~Fyx)]

Question 14

select the best translation into predicate logic, using the following translation key:
Cx: x is a cheetah
Lx: x is a lion
Tx: x is a tiger
Fxy: x is faster than y
Lxy: x is larger than y
-Some lions are larger than all cheetahs.

A) (∃x)[Lx • (∀y)(Lxy ⊃ Cy)]
B) (∃x)(∀y)[(Lx • Lxy) ⊃ Cy]
C) (∃x)[Lx • (∀y)(Cy ⊃ Lxy)]
D) (∀x)[Cx ⊃ (∃y)(Ly • Lxy)]
E) (∃x)[Lx ⊃ (∀y)(Cy ⊃ Lxy)]

Question 15

select the best translation into predicate logic, using the following translation key:
Cx: x is a cheetah
Lx: x is a lion
Tx: x is a tiger
Fxy: x is faster than y
Lxy: x is larger than y
-Some tigers and all cheetahs are faster than all lions.

A) (∃x)[Tx • (∀y)(Ly ⊃ Fxy)] • (∀x)[Cx • (∀y)(Lz ⊃ Fxy)]
B) (∃x)(∀y)(∀z){(Fyz • Fxz) ⊃ [(Lz • Tx) • Cy]}
C) (∃x)(∀y)(∀z){[(Lz • Tx) • Cy] ⊃ (Fyz • Fxz)]}
D) (∃x)(∀y){(Tx • Cy) ⊃ (∀z)[Lz ⊃ (Fzx • Fzy)]}
E) (∀x){Lx ⊃ [(∃y)(Ty • Fyx) • (∀z)(Cz ⊃ Fzx)]}

Question 16

select the best translation into predicate logic, using the following translation key:
Cx: x is a cheetah
Lx: x is a lion
Tx: x is a tiger
Fxy: x is faster than y
Lxy: x is larger than y
-All lions and tigers are larger than some cheetahs, but not faster than all cheetahs.

A) (∀x)(∀y){(Lx • Ty) ⊃ (∃z)[(Cz • Lxz) • Lyz]} • ~(∀x)(∀y)(∀z)(Fxz • Fyz)
B) ~(∃x){Cx • (∀y)[(Ly $\lor$ Ty) ⊃ (Fxy • Lyx)]}
C) (∀x){(Lx $\lor$ Tx) ⊃ ~(∃y)[Cy • (Fyx • Lxy)]}
D) (∃x){Cx ⊃ (∀y)(∀z)[(Ly . Tz) ⊃ (~Fxy . ~Lzx)]}
E) (∀x){(Lx $\lor$ Tx) ⊃ [(∃y)(Cy • Lxy) • ∼(∀y)(Cy ⊃ Fxy)]}

Question 17

select the best translation into predicate logic, using the following translation key:
Cx: x is a cheetah
Lx: x is a lion
Tx: x is a tiger
Fxy: x is faster than y
Lxy: x is larger than y
-Some cheetahs that are not larger than any lion are faster than some tigers.

A) (∃x){[Cx • (∀y)(Ty ⊃ ~Lxy)] • (∃y)(Ly • Fxy)}
B) (∃x){[Cx • (∀y)(Cy • ~Lxy)] ⊃ (∃y)(Ty • Fxy)}
C) (∃x){Cx • (∀y)(∃z)[(Ly • Tx) • (~Lxy • Fxy)]}
D) (∃x){[Cx • (∀y)(~Lxy ⊃ Lx)] • (∃y)(Fxy • Tx)}
E) (∃x){[Cx • (∀y)(Ly ⊃ ∼Lxy)] • (∃y)(Ty • Fxy)}

Question 18

select the best translation into predicate logic, using the following translation key:
Cx: x is a cheetah
Lx: x is a lion
Tx: x is a tiger
Fxy: x is faster than y
Lxy: x is larger than y
-Any lion larger than all cheetahs is not faster than any tiger.

A) (∀x){[Lx ⊃ (∀y)(Lxy ⊃ Cx)] ⊃ ~(∃y)(Ty • Fxy)}
B) (∀x){Tx ⊃ ~(∃y)[(Ly • Lxy) • (∀z)(Cz ⊃ Fxz)]}
C) (∀x){[Lx • (∀y)(Cx ⊃ Lxy)] ⊃ ∼(∃y)(Ty • Fxy)}
D) (∀x){[Lx ⊃ (∀y)(Cy ⊃ Lxy)] ⊃ (∀y)(Ty ⊃ ~Fxy)}
E) (∀x){[Lx • (∀y)(Cx • Lyx)] ⊃ ~(∃y)(Ty ⊃ ~Fxy)}

Question 19

select the best translation into predicate logic, using the following translation key:
Cx: x is a cheetah
Lx: x is a lion
Tx: x is a tiger
Fxy: x is faster than y
Lxy: x is larger than y
-No cheetahs that are larger than some tiger are faster than all lions or tigers.

A) ~(∃x){[Cx • (∃y)(Ty • Lxy)] • (∀y)[(Ly • Ty) ⊃ Fxy]}
B) ~(∃x){[Cx • (∃y)(Ty • Fxy)] • (∀y)[(Ly • Ty) ⊃ Lxy]}
C) (∀x){[Cx • (∃y)(Ty • Lxy)] ⊃ ∼(∀y)[(Ly $\lor$ Ty) ⊃ Fxy]}
D) ~(∃x){[Cx ⊃ (∃y)(Ty • Lxy)] • (∀y)((Ly $\lor$ Ty) ⊃ Fxy]}
E) ~(∃x){[Cx • (∃y)(Ty • Lxy)] ⊃ ~(∀y)(Ly $\lor$ Ty) ⊃ Fxy]}

Question 20

select the best translation into predicate logic, using the following translation key:
Cx: x is a cheetah
Lx: x is a lion
Tx: x is a tiger
Fxy: x is faster than y
Lxy: x is larger than y
-Some lion is faster than all tigers if, and only if, some cheetah is larger than any tiger faster than it.

A) (∃x)[Lx • (∀y)(Ty ⊃ Fxy)] ≡ (∃x){Cx • (∀y)[(Ty • Fyx) ⊃ Lxy]}
B) (∃x)[Lx • (∀y)(Fxy ⊃ Ty)] ≡ (∃x){Cx • (∀y)[Lxy ⊃ (Ty • Fyx)]}
C) (∃x)(∃y)(∀z){[(Lx • Cy) • Tz] • [Fxz ≡ (Fzy ⊃ Lyz)]}
D) (∃x)(∃y)(∀z){[(Lx • Cy) • Tz] • [Fxz ≡ (Lyz ⊃ Fzy)]}
E) (∃x)(∃y)(∀z){[Tz ⊃ (Lx • Cy)] ≡ [(Fxz • Lyz) • Fzy]}

Question 21

select the best English interpretation of the given statements of predicate logic, using the following translation key:
t: two
Ox: x is odd
Ex: x is even
Nx: x is a number
Gxy: x is greater than y
-(∀x)[Nx ⊃ (Ox ⊃ ∼Ex)]

A) All numbers are odd if they are not even.
B) All numbers are odd or even.
C) No even number is odd.
D) No odd number is even.
E) If a number is odd it is even.

Question 22

select the best English interpretation of the given statements of predicate logic, using the following translation key:
t: two
Ox: x is odd
Ex: x is even
Nx: x is a number
Gxy: x is greater than y
-∼(∃x)(Nx • Gxx)

A) Some numbers are not greater than themselves.
B) Some numbers are not greater than any other number.
C) No number is greater than itself.
D) No number is greater than any other number.
E) Some number is no greater than itself.

Question 23

select the best English interpretation of the given statements of predicate logic, using the following translation key:
t: two
Ox: x is odd
Ex: x is even
Nx: x is a number
Gxy: x is greater than y
-~(∃x)[(Nx • ∼Ex) • Gxt]

A) No number that is not even is greater than two.
B) Some even numbers are not greater than two.
C) Some number greater than two is not even.
D) Two is the smallest even number.
E) Two is not the smallest even number.

Question 24

select the best English interpretation of the given statements of predicate logic, using the following translation key:
t: two
Ox: x is odd
Ex: x is even
Nx: x is a number
Gxy: x is greater than y
-(∀x){(Nx • Ox) ⊃ (∃y)[(Ey • Ny) • Gyx)]}

A) For every odd number, there is an even number greater than it.
B) Every odd number is greater than some even number.
C) All odd numbers are smaller than all even numbers.
D) Any number smaller than every odd number is even.
E) Any number greater than every odd number is even.

Question 25

select the best English interpretation of the given statements of predicate logic, using the following translation key:
t: two
Ox: x is odd
Ex: x is even
Nx: x is a number
Gxy: x is greater than y
-(∀x){(Ex • Nx) ⊃ (∀y)[(Oy • Ny) ⊃ ∼Gxy]}

A) All even numbers are smaller than all odd numbers.
B) No even number is greater than any odd number.
C) No odd number is greater than any even number.
D) All even numbers are greater than all odd numbers.
E) No even number is greater than every odd number.

Question 26

consider the following domain, assignment of objects in the domain, and assignments sets to predicates.
Domain = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto}
a = Mercury b = Jupiter c = Saturn d = Pluto
Mx = {Mercury, Mars}
Px = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}
Cxy = {, , , , , , , , , ,
, , , , , , , , , , , , , , , , , , , , , , , , , }
-Given the customary truth tables, which of the following theories is modeled by the above interpretation?

A) Pa • Pd
Cad • ~Cda
B) Pb • Pd
Cbd • ~Cdb
C) Pa • Pb
Cab • ∼Cba
D) Pb • Pa
Cba • ~Cab
E) Pc • Pb
Ccb • ~Cbc

Question 27

consider the following domain, assignment of objects in the domain, and assignments sets to predicates.
Domain = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto}
a = Mercury b = Jupiter c = Saturn d = Pluto
Mx = {Mercury, Mars}
Px = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}
Cxy = {, , , , , , , , , ,
, , , , , , , , , , , , , , , , , , , , , , , , , }
-Given the customary truth tables, which of the following theories is modeled by the above interpretation?

A) (Ma • Pb) • (Pc $\lor$ Pd) (∃x)Cxa • ~(∃x)Cxd
B) (Pa • Mb) • (Pc $\lor$ Pd) (∃x)Cxd $\lor$ ~(∃x)Cxa
C) (Pa • Pb) • (Pc $\lor$ Pd)
(∃x)Cxd • ∼(∃x)Cxa
D) (Pa • Pb) • (Mc $\lor$ Pd) (∃x)Cxd $\lor$ ~(∃x)Cxa
E) (Pa • Pb) • (Pc $\lor$ Md) (∃x)Cxa • ~(∃x)Cxd

Question 28

consider the following domain, assignment of objects in the domain, and assignments sets to predicates.
Domain = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto}
a = Mercury b = Jupiter c = Saturn d = Pluto
Mx = {Mercury, Mars}
Px = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}
Cxy = {, , , , , , , , , ,
, , , , , , , , , , , , , , , , , , , , , , , , , }
-Given the customary truth tables, which of the following theories is modeled by the above interpretation?

A) Ma • ~Mc
(∃x)Cxc
∼(∃x)(∼Cxc • Mx)
B) Ma • Mc (∃x)Ccx (∃x)(~Cxc • Mx)
C) ~(Ma • Mc)
(∃x)Ccx
~(∃x)(Cxc • Mx)
D) ~(Ma $\lor$ Mc) (∃x)Ccx
~(∃x)(Ccx • Mx)
E) Ma ≡ ~Mc
(∃x)Cax
~(∃x)(Cxc • ~Mx)

Question 29

consider the following domain, assignment of objects in the domain, and assignments sets to predicates.
Domain = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto}
a = Mercury b = Jupiter c = Saturn d = Pluto
Mx = {Mercury, Mars}
Px = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}
Cxy = {, , , , , , , , , ,
, , , , , , , , , , , , , , , , , , , , , , , , , }
-Given the customary truth tables, which of the following theories is modeled by the above interpretation?

A) (∀x){Px ⊃ [(∃y)Cxy • (∃y)Cyx]} Cab • ~Cba
Cca • ~Cac Cad • ~Cda
B) (∀x){Px ⊃ [(∃y)Cxy ≡ (∃x)Cyx]} Cab • ~Cba
Cac • ~Cca Cad • ~Cda
C) (∀x)[Px ⊃ (∃y)(Cxy • Cyx)] Cba • ~Caa
Cca • ~Ccc Cda • ~Cdd
D) (∀x){Px ⊃ [(∃y)Cxy $\lor$ (∃y)Cyx]}
Cab • ∼Cba Cac • ∼Cca Cad • ∼Cda
E) (∀x)[Px ⊃ (∃y)(Cxy $\lor$ Cyx)]
Cab • ~Cba Cac • Cad
~(~Cca • ~Cda)

Question 30

consider the following domain, assignment of objects in the domain, and assignments sets to predicates.
Domain = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto}
a = Mercury b = Jupiter c = Saturn d = Pluto
Mx = {Mercury, Mars}
Px = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}
Cxy = {, , , , , , , , , ,
, , , , , , , , , , , , , , , , , , , , , , , , , }
-Given the customary truth tables, which of the following theories is modeled by the above interpretation?

A) (∀x)(Px ⊃ Mx)
~(∃x)Px
(∀x)(∀y)[(Px • Py) ⊃ Cxy]
B) (∀x)(Mx ≡ Px)
~(∀x)Mx (∀x)[(∀y)Cxy ⊃ Py]
C) (∀x)(Mx $\lor$ Px)
~(∃x)Mx
(∀x)[Px ⊃ ~(∀y)(Py ⊃ Cxy)]
D) (∀x)(~Mx $\lor$ Px)
~(∀x)(Px
(∀x)[Px ⊃ (∀y)(Py ⊃ Cxy)]
E) (∀x)(Mx ⊃ Px)
∼(∀x)Px
(∀x)[∼Px ⊃ (∀y)(Py ⊃ Cyx)]

Question 31

consider the following domain, assignment of objects in the domain, and assignments sets to predicates.
Domain = {1, 2, 3, ..., 28, 29, 30}
a = 1
e = 21
b = 2
f = 23
c = 4
g = 27
d = 20
h = 29
Ex = {2, 4, 6, ..., 28, 30}
Ox = {1, 3, 5, ..., 27, 29}
Px = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
Sxyz = The set of all triples such that the first is the sum of the second and third
{<2, 1, 1>, <3, 1, 2>, <3, 2, 1>, <4, 1, 3>, <4, 2, 2>, <4, 3, 1>, <5, 1, 4>, ... }
-Given the customary truth tables, which of the following theories is modeled by the above interpretation?

A) Sdca • Shgb
(∃x)(∃y)Sdxy • ~(∃x)(∃y)Sgxy
B) Scbb • Seda
(∃x)(∃y)Scxy • ~(∃x)(∃y)Sbxy
C) Secb • Shga
(∃x)(∃y)Sgxy • ~(∀x)(∀y)Sfxy
D) Sfeb • Sgfc
(∃x)(∃y)Shxy • ∼(∀x)(∀y)Shxy
E) Shgb • Sfbe
(∃x)(∃y)Sgxy • ~(∃x)Sxcd

Question 32

consider the following domain, assignment of objects in the domain, and assignments sets to predicates.
Domain = {1, 2, 3, ..., 28, 29, 30}
a = 1
e = 21
b = 2
f = 23
c = 4
g = 27
d = 20
h = 29
Ex = {2, 4, 6, ..., 28, 30}
Ox = {1, 3, 5, ..., 27, 29}
Px = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
Sxyz = The set of all triples such that the first is the sum of the second and third
{<2, 1, 1>, <3, 1, 2>, <3, 2, 1>, <4, 1, 3>, <4, 2, 2>, <4, 3, 1>, <5, 1, 4>, ... }
-Given the customary truth tables, which of the following theories is modeled by the above interpretation?

A) (∃x)(Ex • Sfex)
(∀x)(Ox ⊃ ∼Sfex) (∃x)(∃y)(Oy • Sxey)
B) (∃x)(Ex • Sefx) (∀x)(Ox ⊃ ~Sefx) (∃x)(∃y)(Oy • Sxfy)
C) (∃x)(Ox • Sfex) (∀x)(Ex ⊃ ~Sfex) (∃x)(∃y)(Ey • Sxey)
D) (∃x)(Ox • Sefx) (∀x)(Ex ⊃ ~Sefx) (∃x)(∃y)(Ey • Sxfy)
E) (∃x)(Ox • Sfxe) (∀x)(Ex ⊃ ~Sfxe) (∃x)(∃y)(Ey • Sfxy)

Question 33

consider the following domain, assignment of objects in the domain, and assignments sets to predicates.
Domain = {1, 2, 3, ..., 28, 29, 30}
a = 1
e = 21
b = 2
f = 23
c = 4
g = 27
d = 20
h = 29
Ex = {2, 4, 6, ..., 28, 30}
Ox = {1, 3, 5, ..., 27, 29}
Px = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
Sxyz = The set of all triples such that the first is the sum of the second and third
{<2, 1, 1>, <3, 1, 2>, <3, 2, 1>, <4, 1, 3>, <4, 2, 2>, <4, 3, 1>, <5, 1, 4>, ... }
-Given the customary truth tables, which of the following theories is modeled by the above interpretation?

A) Sgfc • Sgcf
(∀x)(∀y)(∀z)(Sxyz ≡ Sxzy)
B) Sgba • Sgab
(∀x)(∀y)[(∀z)Sxyz ≡ (∀z)Sxzy]
C) Seda • Sead
(∀x)(∀y)(∀z)(Sxyz ≡ Szyx)
D) Sfea • Sfae
(∀x)(∀y)(∀z)(Sxyz ≡ Syxz)
E) Sdba • Sdab
(∀x)(∀y)[(∀z)(Sxyz ≡ (∀z)Syxz]

Question 34

consider the following domain, assignment of objects in the domain, and assignments sets to predicates.
Domain = {1, 2, 3, ..., 28, 29, 30}
a = 1
e = 21
b = 2
f = 23
c = 4
g = 27
d = 20
h = 29
Ex = {2, 4, 6, ..., 28, 30}
Ox = {1, 3, 5, ..., 27, 29}
Px = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
Sxyz = The set of all triples such that the first is the sum of the second and third
{<2, 1, 1>, <3, 1, 2>, <3, 2, 1>, <4, 1, 3>, <4, 2, 2>, <4, 3, 1>, <5, 1, 4>, ... }
-Given the customary truth tables, which of the following theories is modeled by the above interpretation?

A) (∃x)(∃y)[(Ex • Ey) • Sfxy] (∃x)(∃y)[(Ox • Oy) • Sfxy]
(∀x){Ex ⊃ {(∃y)(∃z)[(Ey • Ez) • Sxyz)] $\lor$ (∃y)(∃z)[(Ox • Oy) • Sxyz]}}
B) (∃x)(∃y)[(Ex • Ey) • Sdxy]
(∃x)(∃y)[(Ox • Oy) • Sdxy]
(∀x){Ex ⊃ {(∃y)(∃z)[(Ey • Ez) • Sxyz)] $\lor$ (∃y)(∃z)[(Ox • Oy) • Sxyz]}}
C) (∃x)(∃y)[(Ex • Ey) • Sdxy]
(∃x)(∃y)[(Ox • Oy) • Sdxy]
(∀x){Ex ⊃ {(∃y)(∃z)[(Ey • Ez) • Sxyz)] • (∃y)(∃z)[(Ox • Oy) • Sxyz]}}
D) (∃x)(∃y)[(Ex • Ey) • Sfxy]
(∃x)(∃y)[(Ox • Oy) • Sfxy]
(∀x){Ex ⊃ {(∃y)(∃z)[(Ey • Ez) • Sxyz)] • (∃y)(∃z)[(Ox • Oy) • Sxyz]}}
E) (∃x)(∃y)[(Ex • Ey) • Sdxy]
(∃x)(∃y)[(Ox • Oy) • Sdxy]
(∀x){Ox ⊃ {(∃y)(∃z)[(Ey • Ez) • Sxyz)] $\lor$ (∃y)(∃z)[(Ox • Oy) • Sxyz]}}

Question 35

consider the following domain, assignment of objects in the domain, and assignments sets to predicates.
Domain = {1, 2, 3, ..., 28, 29, 30}
a = 1
e = 21
b = 2
f = 23
c = 4
g = 27
d = 20
h = 29
Ex = {2, 4, 6, ..., 28, 30}
Ox = {1, 3, 5, ..., 27, 29}
Px = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
Sxyz = The set of all triples such that the first is the sum of the second and third
{<2, 1, 1>, <3, 1, 2>, <3, 2, 1>, <4, 1, 3>, <4, 2, 2>, <4, 3, 1>, <5, 1, 4>, ... }
-Given the customary truth tables, which of the following theories is modeled by the above interpretation?

A) (∃x)(∀y)(∀z)Sxyz (∃x)(∀y)(∀z)Syxz
B) (∃x)(∀y)(∀z)∼Sxyz (∃x)(∀y)(∀z)Syxz
C) (∃x)(∀y)(∀z)Sxyz (∃x)(∀y)(∀z)∼Syxz
D) (∃x)(∀y)∼Sxyy
(∃x)(∀y)Syxy
E) (∃x)(∀y)(∀z)∼Sxyz (∃x)(∀y)(∀z)∼Syxz

Question 36

provide a conterexample in a finite domain to each given invalid argument.
-1. Ad ⊃ (∀x)Fdx / (∃x)Fdx

A) Counterexample in a domain of one member, in which: Ad: True Fdd: False
B) Counterexample in a domain of one member, in which: Ad: False Fdd: False
C) Counterexample in a domain of one member, in which: Ad: True Fdd: True
D) Counterexample in a domain of two members, in which: Ad: True Fdd: False
Ab: True Fdb: False
E) Counterexample in a domain of two members, in which: Ad: False Fdd: True
Ab: False Fdb: False

Question 37

provide a conterexample in a finite domain to each given invalid argument.
-1. (∃x)(Dxa • Ex)
2) (∃x)(Dxa • Fx) / (∃x)(Ex • Fx)

A) Counterexample in a domain of one member, in which: Ea: True Fa: False Daa: True
B) Counterexample in a domain of one member, in which: Ea: False Fa: True Daa: False
C) Counterexample in a domain of two members, in which: Ea: True Fa: True Daa: True
Eb: True Fb: False Dba: False
D) Counterexample in a domain of two members, in which: Ea: True Fa: False Daa: False
Eb: False Fb: False Dba: True
E) Counterexample in a domain of two members, in which: Ea: True Fa: False Daa: True
Eb: False Fb: True Dba: True

Question 38

provide a conterexample in a finite domain to each given invalid argument.
-1. (∀x)[Hx ⊃ (∃y)(Ix • Jxy)]
2) Ha
3) Ib / Jab

A) Counterexample in a domain of two members, in which:
Ha: True Ia: True Jaa: True Jba: False
Hb: True Ib: False Jab: False Jbb: True
B) Counterexample in a domain of two members, in which:
Ha: True Ia: False Jaa: True Jba: False
Hb: False Ib: True Jab: False Jbb: True
C) Counterexample in a domain of two members, in which:
Ha: True Ia: True Jaa: True Jba: True
Hb: False Ib: True Jab: False Jbb: True
D) Counterexample in a domain of two members, in which:
Ha: False Ia: True Jaa: False Jba: True
Hb: True Ib: True Jab: True Jbb: True
E) Counterexample in a domain of two members, in which:
Ha: True Ia: True Jaa: False Jba: False
Hb: True Ib: False Jab: True Jbb: True

Question 39

provide a conterexample in a finite domain to each given invalid argument.
-1. (∀x)[Px ⊃ (∃y)Qxy]
2) (∀x)[(∃y)Qxy ⊃ (∃y)Rxy]
3) Pa / Rab

A) Counterexample in a domain of two members, in which:
Pa: True Qaa: False Qba: True Raa: True Rba: False
Pb: False Qab: False Qbb: True Rab: False Rbb: True
B) Counterexample in a domain of two members, in which:
Pa: True Qaa: True Qba: False Raa: True Rba: True
Pb: False Qab: True Qbb: False Rab: False Rbb: False
C) Counterexample in a domain of two members, in which:
Pa: True Qaa: False Qba: False Raa: False Rba: True
Pb: True Qab: True Qbb: False Rab: False Rbb: True
D) Counterexample in a domain of two members, in which:
Pa: True Qaa: True Qba: True Raa: True Rba: False
Pb: False Qab: False Qbb: True Rab: True Rbb: True
E) Counterexample in a domain of two members, in which:
Pa: True Qaa: True Qba: True Raa: False Rba: False
Pb: True Qab: False Qbb: False Rab: False Rbb: False

Question 40

provide a conterexample in a finite domain to each given invalid argument.
-1. (∀x)[(∃y)(Fxy • Gy) ⊃ (∀y)(Gy ⊃ Fxy)]
2) Ga / (∀x)(Fxa ⊃ Fax)

A) Counterexample in a domain of two members, in which: Ga: True Faa: True Fba: True
Gb: False Fab: False Fbb: True
B) Counterexample in a domain of two members, in which: Ga: True Faa: False Fba: True
Gb: True Fab: False Fbb: False
C) Counterexample in a domain of two members, in which: Ga: True Faa: True Fba: True
Gb: False Fab: True Fbb: False
D) Counterexample in a domain of two members, in which: Ga: False Faa: True Fba: False
Gb: False Fab: True Fbb: True
E) Counterexample in a domain of two members, in which: Ga: True Faa: False Fba: True
Gb: True Fab: True Fbb: False

Question 41

1. (∀x)[Ex ⊃ (∀y)(Fy • Gxy)]
2. (∃x)(Ex • Hxb)
-Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

A) Eb ⊃ (∀y)(Fy • Gxy)
B) Ea • Hab
C) Ea ⊃ (∀y)(Fy • Gby)
D) Hbb
E) (∀x)[Ex ⊃ (Fx • Gxx)]

Question 42

1. (∀x)[Ex ⊃ (∀y)(Fy • Gxy)]
2. (∃x)(Ex • Hxb)
-Which of the following propositions is derivable from the given premises in F?

A) ~Fb
B) (∀x)Gax
C) (∀x)Gbx
D) (∃x)(∀y)(Gxy ⊃ Hxy)
E) (∃x)(∃y)(Gxy • Hxy)

Question 43

1. (∀x)(∃y)Axy ⊃ (∀x)(∃y)Bxy
2. (∃x)(∀y)∼Bxy
-Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

A) (∃y)Axy ⊃ (∀x)(∃y)Bxy
B) ~Bxy
C) (∃x)∼(∃y)Bxy
D) (∀x)(∃y)Bxy
E) (∀y)~Byy

Question 44

1. (∀x)(∃y)Axy ⊃ (∀x)(∃y)Bxy
2. (∃x)(∀y)∼Bxy
-Which of the following propositions is derivable from the given premises in F?

A) (∃x)(∀y)∼Axy
B) (∃x)(∀y)~Ayx
C) (∀x)(∃y)Axy
D) (∀x)(∃y)Ayx
E) (∃x)(∃y)(Axy • Bxy)

Question 45

1. (∃x)[Dx • (∀y)(Ey ⊃ Fxy)]
2. (∀x)(Dx ⊃ Ex)
-Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

A) Dx ⊃ Ea
B) Dx • (∀y)(Ey ⊃ Fyy)
C) Da • (∀y)(Ey ⊃ Fay)
D) Da
E) Da • (∀y)(Ea ⊃ Fay)

Question 46

1. (∃x)[Dx • (∀y)(Ey ⊃ Fxy)]
2. (∀x)(Dx ⊃ Ex)
-Which of the following propositions is derivable from the given premises in F?

A) (∀x)(∀y)[(Dx • Dy) ⊃ Fxy]
B) (∀x)(Dx ⊃ Fxx)
C) (∀x)[Ex ⊃ (∃y)Fxy]
D) (∀x)[Ex ⊃ (∀y)Fyx]
E) (∃y)(Ey • Fyy)

Question 47

1. (∀x)[(Cx • Exa) ⊃ Dx]
2. Cd • ∼Dd
-Which of the following propositions is an immediate (one-step) consequence in F of the given premise?

A) Cd • Eda
B) Ca • Eaa
C) Eda ⊃ Dd
D) ∼Dd • Cd
E) (Cx • Exd) ⊃ Dd

Question 48

1. (∀x)[(Cx • Exa) ⊃ Dx]
2. Cd • ∼Dd
-Which of the following propositions is derivable from the given premises in F?

A) ∼Eda
B) ~(∃x)Exa
C) (∀x)Exa
D) Eda
E) Eaa

Question 49

1. (∀x)[(Px $\lor$ Qx)] ⊃ Rxx]
2. (∀x){Qx ⊃ [(∃y)Rxy ⊃ Sxx]}
3. Pn • Qn
-Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

A) Qn
B) (Px $\lor$ Qx) ⊃ Rnn
C) Qn ⊃ [(∃y)Rny ⊃ Sxx]
D) (∀x)[Qx ⊃ (Rxy ⊃ Sxx)]
E) Pn

Question 50

1. (∀x)[(Px $\lor$ Qx)] ⊃ Rxx]
2. (∀x){Qx ⊃ [(∃y)Rxy ⊃ Sxx]}
3. Pn • Qn
-Which of the following propositions is derivable from the given premises in F?

A) Rnn • Snn
B) (∀x)Rnx
C) (∀x)Rxn
D) (∀x)(Px ⊃ Sxx)
E) Rnn ≡ ~Snn

Question 51

1. (∀x)[Ax ⊃ (∃y)(By • Cxy)]
2. (∃x)(Ax • Dx)
3. (∀x)(Bx ⊃ Ex)
-Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

A) Ba ⊃ Eb
B) Ax ⊃ (∃y)(By • Cyy)
C) Ax • Dy
D) Ae • De
E) Bb ⊃ Ex

Question 52

1. (∀x)[Ax ⊃ (∃y)(By • Cxy)]
2. (∃x)(Ax • Dx)
3. (∀x)(Bx ⊃ Ex)
-Which of the following propositions is derivable from the given premises in F?

A) (∃x)[Dx • (∃y)(Ey • Cxy)]
B) (∃x)[(Dx • Ex) • Cxx]
C) (∃x)[Dx • (∃y)(Ey • Cyx)]
D) (∃x)[Dx • (∀y)(Ey ⊃ Cyx)]
E) (∃x)[Dx • (∀y)(Ey ⊃ Cxy)]

Question 53

1. (∀x)[Ax ⊃ (∀y)(By ⊃ Cxy)]
2. (∃x)[Ex • (∀y)(Hy ⊃ Cxy)]
3. (∀x)(∀y)(∀z)[(Cxy • Cyz) ⊃ Cxz]
4. (∀x)(Ex ⊃ Bx)
-Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

A) Ax ⊃ (By ⊃ Cxy)
B) (∀z)[(Cab • Cbz) ⊃ Caz]
C) Eb ⊃ Be
D) Eb • (∀y)(Hy ⊃ Cby)
E) Ax ⊃ (∀y)(By ⊃ Cby)

Question 54

1. (∀x)[Ax ⊃ (∀y)(By ⊃ Cxy)]
2. (∃x)[Ex • (∀y)(Hy ⊃ Cxy)]
3. (∀x)(∀y)(∀z)[(Cxy • Cyz) ⊃ Cxz]
4. (∀x)(Ex ⊃ Bx)
-Which of the following propositions is derivable from the given premises in F?

A) (∀x){Ex ⊃ [(∃y)Cxy ⊃ (∃y)Cyx]}
B) (∀x)[Bx ⊃ (Ex $\lor$ Ax)]
C) (∀x)[Ax ⊃ (∀y)(By ⊃ Cyx)]
D) (∀x)[Ax ⊃ (∀y)(Hy ⊃ Cxy)]
E) (∀x){Ax ⊃ (∀y)[(Hy • By) ≡ Cxy]}

Question 55

(∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∀x)(∀y)∼Rxy ⊃ ∼(∃x)Px]
-Consider assuming '(∀x)[Px ⊃ (∃y)Rxy]' for a conditional proof of the above logical truth. Which of the
Following propositions is a legitimate second step in that proof?

A) Assume '~(∃x)Px' for a nested conditional proof.
B) Assume '(∀x)(∀y)∼Rxy' for a nested conditional proof.
C) Assume '~(∀x)(∀y)~Rxy' for a nested indirect proof.
D) Px ⊃ (∃y)Ryy
E) Py ⊃ (∃y)Ryy

Question 56

(∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∀x)(∀y)∼Rxy ⊃ ∼(∃x)Px]
-Which of the following propositions is also derivable in F?

A) (∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∃x)~Px ⊃ (∃x)(∃y)Rxy]
B) (∀x)[(∃y)Rxy ⊃ ~Px] ⊃ [(∃x)Px ⊃ (∃x)(∃y)Rxy]
C) (∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∃x)Px ⊃ (∃x)(∃y)Rxy]
D) (∀x)[~(∃y)Rxy ⊃ Px] ⊃ [(∃x)Px ⊃ (∃x)(∃y)Rxy]
E) (∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∃x)~Px ⊃ (∃x)(∃y)~Rxy]

Question 57

(∀x)[Px ⊃ (∃y)Qxy] ⊃ [(∃x)Px ⊃ (∃x)(∃y)Qxy]
-Consider assuming '(∀x)[Px ⊃ (∃y)Qxy]' for a conditional proof of the above logical truth. Which of the
Following propositions is a legitimate second step in that proof?

A) Assume '(∃x)Px' for a nested indirect proof.
B) Assume '(∃x)(∃y)Qxy' for a nested indirect proof.
C) Assume '(∃x)Px' for a nested conditional proof.
D) Assume '(∃x)(∃y)Qxy' for a nested conditional proof.
E) Px ⊃ (∃y)Qyy

Question 58

(∀x)[Px ⊃ (∃y)Qxy] ⊃ [(∃x)Px ⊃ (∃x)(∃y)Qxy]
-Which of the following propositions is also derivable in F?

A) (∀x)[~(∃y)Qxy ⊃ ~Px] ⊃ [(∀x)Px $\lor$ ~(∃x)(∃y)Qxy]
B) (∀x)[Px ⊃ (∃y)Qxy] ⊃ [(∀x)∼Px $\lor$ (∃x)(∃y)Qxy]
C) (∀x)[~(∃y)Qxy ⊃ ~Px] ⊃ [~(∀x)Px $\lor$ ~(∃x)(∃y)Qxy]
D) (∀x)[~(∃y)Qxy ⊃ ~Px] ⊃ ~[(∀x)Px $\lor$ ~(∃x)(∃y)Qxy]
E) (∀x)[Px ⊃ (∃y)Qxy] ⊃ [(∀x)∼Px $\lor$ (∃x)(∃y)~Qxy]

Question 59

[(∀x)Pax • ∼(∃x)Pxa] ⊃ (∀x)(Pax • ∼Pxa)
-Consider assuming '(∀x)Pax • ∼(∃x)Pxa' for a conditional proof of the above logical truth. Which of the
Following propositions is a legitimate second step in that proof?

A) Assume '(∀x)Pax' for a nested indirect proof.
B) Assume ~(∀x)Pax' for a nested indirect proof.
C) ~(∃x)Pxa
D) (∀x)~Pxa
E) ∼(∃x)Pxa • (∀x)Pax

Question 60

[(∀x)Pax • ∼(∃x)Pxa] ⊃ (∀x)(Pax • ∼Pxa)
-Which of the following propositions is also derivable in F?

A) [(∃x)∼Pax $\lor$ ~(∃x)Pxa] $\lor$ (∀x)(Pax • ∼Pxa)
B) [(∃x)∼Pax $\lor$ (∃x)Pxa] $\lor$ (∀x)(Pax • ∼Pxa)
C) [(∃x)Pax $\lor$ (∃x)Pxa] $\lor$ (∀x)(Pax • Pxa)
D) [(∃x)Pax $\lor$ ~(∃x)Pxa] $\lor$ (∀x)(Pax • ∼Pxa)
E) [(∃x)∼Pax • (∃x)Pxa] • (∀x)(Pax • ∼Pxa)

Question 61

All cheetahs are faster than some tigers. Everything is striped if, and only if, it is a tiger. So, if some things are cheetahs, then some things have stripes. (Cx: x is a cheetah; Tx: x is a tiger; Sx: x has stripes; Fxy: x is faster than y)
-Which of the following is the best translation into F of this argument?

A) 1. (∀x)[Cx ⊃ (∃y)(Ty • Fyx)] 2. (∀x)[(Sx ⊃ Tx) • (Tx ⊃ Sx)] / (∃x)Cx ⊃ (∃x)Sx
B) 1. (∀x)(∃y)[Cx • (Ty • Fxy)] 2. (∀x)Sx ≡ (∀x)Tx / (∃x)Cx ⊃ (∃x)Sx
C) 1. (∀x)[Cx ⊃ (∃y)(Ty • Fxy)] 2. (∀y)(Sy ≡ Ty) / (∃x)Cx ⊃ (∃x)Sx
D) 1. (∀x)[Cx ⊃ (∃y)(Ty • Fyx)] 2. (∀x)(Sx ≡ Tx) / (∃x)(Cx ⊃ Sx)
E) 1. (∀x)[(∀y)(Fxy • Ty) ⊃ Cx] 2. (∀x)(Sx ⊃ Tx) • (∀x)(Tx ⊃ Sx) / (∃x)(Cx ⊃ Sx)

Question 62

All cheetahs are faster than some tigers. Everything is striped if, and only if, it is a tiger. So, if some things are cheetahs, then some things have stripes. (Cx: x is a cheetah; Tx: x is a tiger; Sx: x has stripes; Fxy: x is faster than y)
-Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

A) Sx ≡ Ty
B) Sx ≡ (∀x)Tx
C) Ca ⊃ (∃y)(Ty • Fya)
D) Sx ⊃ Tx
E) Cx ⊃ (∃y)(Ty • Fxy)

Question 63

All cheetahs are faster than some tigers. Everything is striped if, and only if, it is a tiger. So, if some things are cheetahs, then some things have stripes. (Cx: x is a cheetah; Tx: x is a tiger; Sx: x has stripes; Fxy: x is faster than y)
-Which of the following claims can also be derived from the premises of this argument?

A) (∀x)∼Sx ⊃ (∀x)∼Cx
B) (∀x)(∃y)[Fxy ⊃ (Cy • Tx)]
C) (∃x)(Cx • Tx)
D) ~(∃x)(Cx • Tx)
E) (∀x)(∃y)[Fxy ⊃ (Cx • Ty)]

Question 64

All philosophers respect each other. Some philosopher doesn't study some philosopher. Anything which respects something without studying it is open-minded, if ignorant. So something is open-minded and ignorant. (Ix: x is ignorant; Ox: x is open-minded; Px: x is a philosoper; Rxy: x respects y; Sxy: x studies y)
-Which of the following is the best translation into F of this argument?

A) 1. (∀x)(∀y)[(Px • Py) ⊃ (Rxx • Ryy)] 2. (∃x)(∃y)[(Px • Py) • ~Sxy]
3) (∀x)(∀y)[(Rxy • ~Sxy) ⊃ (Ox $\lor$ Ix)] / (∃x)(Ox • Ix)
B) 1. (∀x)(∀y)[(Px • Py) ⊃ (Rxy ≡ Ryx)] 2. (∃x)(∃y)[(Px • Py) • ~Sxy]
3) (∀x)(∀y)[(Rxy • ~Sxy) ⊃ (Ox $\lor$ Ix)] / (∃x)(Ox • Ix)
C) 1. (∀x)(∀y)[(Px • Py) ⊃ (Rxy • Ryx)] 2. (∃x)(∃y)[(Px • Py) • ~Sxy]
3) (∀x)(∀y)[(Rxy • ~Sxy) ⊃ (Ox $\lor$ Ix)] / (∃x)(Ox • Ix)
D) 1. (∀x)(∀y)[(Px • Py) ⊃ (Rxy • Ryx)] 2. (∃x)[Px • (∃y)(Py • ∼Sxy)]
3) (∀x)(∀y)[(Rxy • ∼Sxy) ⊃ (Ox • Ix)] / (∃x)(Ox • Ix)
E) 1. (∀x)(∀y)[(Px • Py) ⊃ (Rxx • Ryy)] 2. (∃x)(∃y)[(Px • Py) • ~Sxy]
3) (∀x)(∀y)[(Rxy • ~Sxy) ⊃ (Ox $\lor$ Ix)] / (∃x)(Ox $\lor$ Ix)

Question 65

All philosophers respect each other. Some philosopher doesn't study some philosopher. Anything which respects something without studying it is open-minded, if ignorant. So something is open-minded and ignorant. (Ix: x is ignorant; Ox: x is open-minded; Px: x is a philosoper; Rxy: x respects y; Sxy: x studies y)
-Which of the following propositions is an immediate (one-step) consequence in F of the
Given premises?

A) Ox • Ix
B) (∀y)[(Px • Py) ⊃ (Ryy • Ryy)]
C) (∃y)[(Px • Py) ~ Sxy]
D) Pa • (∃y)(Py • ∼Say)
E) (∀y)[(Rxy • ~Sxy) ⊃ (Ox $\lor$ Ix)]

Question 66

All philosophers respect each other. Some philosopher doesn't study some philosopher. Anything which respects something without studying it is open-minded, if ignorant. So something is open-minded and ignorant. (Ix: x is ignorant; Ox: x is open-minded; Px: x is a philosoper; Rxy: x respects y; Sxy: x studies y)
-Which of the following claims can also be derived from the premises of this argument?

A) Rab • ∼Sab
B) ~Rab • ~Sab
C) ~Rba • ~Sab
D) Rba • Sab
E) Rba • Sbb

Question 67

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
-1. (∀x)(Ax ⊃ Dex)
2) (∃x)(Bx • Dxe)
3) (∀x)(Bx ⊃ Ax) / (∃x)[Ax • (Dxe • Dex)]

A) Valid
B) Invalid. Counterexample in a domain of one member, in which: Ae: True Be: False Dee: False
C) Invalid. Counterexample in a domain of two members, in which:
Aa: True Ba: True Daa: True Dea: False
Ae: True Be: True Dae: False Dee: False
D) Invalid. Counterexample in a domain of two members, in which:
Aa: True Ba: False Daa: True Dea: True
Ae: False Be: False Dae: False Dee: True
E) Invalid. Counterexample in a domain of two members, in which: Aa: False Ba: True Daa: False Dea: False
Ae: True Be: True Dae: True Dee: False

Question 68

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.

-1. (?x)(?y)(Pxy • ?Pyx)
2) (?x)[(?y)Pxy ? (?y)Qxy] / (?x)(?y)(Qxy • ?Pyx)

A) Valid
B) Invalid. Counterexample in a domain of three members, in which:
$\begin{array}{lll}\text { Paa: True } & \text { Pab: True } & \text { Pac: False } \\\text { Pba: False } & \text { Pbb: True } & \text { Pbc: True } \\\text { Pca: True } & \text { Pcb: False } & \text { Pcc: True } \\\text { Qaa: False } & \text { Qab: False } & \text { Qac: True } \\\text { Qba: True } & \text { Qbb:True } & \text { Qbc: True } \\\text { Qca: True } & \text { Qcb:True } & \text { Qcc: True }\end{array}$

C) Invalid. Counterexample in a domain of three members, in which:
$\begin{array}{lll}\text { Paa: True } & \text { Pab: True } & \text { Pac: False } \\\text { Pba: False } & \text { Fbb: False } & \text { Pbc: False } \\\text { Pca: False } & \text { Pcb: False } & \text { Pcc: True } \\\text { Qaa: False } & \text { Qab: False } & \text { Qac: True } \\\text { Qba: True } & \text { Qbb:True } & \text { Qbc: False } \\\text { Qca: True } & \text { Qcb: True } & \text { Qcc: True }\end{array}$

D) Invalid. Counterexample in a domain of three members, in which:
$\begin{array}{lll}\text { Paa: False } & \text { Pab: Ture } & \text { Pac: False } \\\text { Pba: False } & \text { Pbb: True } & \text { Pbc: True } \\\text { Pca: True } & \text { Pcb: True } & \text { Pcc: True } \\\text { Qaa: False } & \text { Qab:True } & \text { Qac: True } \\\text { Qba: False } & \text { Qbb:True } & \text { Qbc: True } \\\text { Qca: True } & \text { Qcb:True } & \text { Qcc: False }\end{array}$

E) Invalid. Counterexample in a domain of three members, in which:
$\begin{array}{lll}\text { Paa: True } & \text { Pab: True } & \text { Pac: False } \\\text { Pla: False } & \text { Pbb: False } & \text { Ploc: True } \\\text { Pca: True } & \text { Pcb: False } & \text { Pcc: True } \\\text { Qaa: False } & \text { Qab: False } & \text { Qac: True } \\\text { Qba: False } & \text { Qbb: False } & \text { Qbc: False } \\\text { Qca: True } & \text { Qcb: True } & \text { Qcc: False }\end{array}$

Question 69

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.

-1. (?x)[Tx ? (?y)(Sy • Wxy)]
2) (?x)(Sx • Vx)
3) (?x)(Tx • Rx) / (?x)[(Rx • Vx) • (?y)Wxy]

A) Valid
B) Invalid. Counterexample in a domain of one member, in which: Ta: True Sa: False Waa: True
Ra: False Va: True
C) Invalid. Counterexample in a domain of two members, in which:
$\begin{array}{lll}\text { Ta: False } & \text { Sa: False } & \text { Waa: True } \\\text { Tb: True } & \text { Sb: True } & \text { Wab: False } \\\text { Ra: True } & \text { Va: False } & \text { Wba: False } \\\text { Rb: False } & \text { Vb: False } & \text { Wbb:True }\end{array}$

D) Invalid. Counterexample in a domain of two members, in which:
$\begin{array}{lll}\text { Ta: False } & \text { Sa: True } & \text { Waa: True } \\\text { Tb: True } & \text { Sb: True } & \text { Wab: True } \\\text { Ra: False } & \text { Va: False } & \text { Wba: False } \\\text { Rb: False } & \text { Vb:True } & \text { Wbb: False }\end{array}$

E) Invalid. Counterexample in a domain of two members, in which:
$\begin{array}{llll}\text { Ta: True } & \text { Sa: True } & \text { Waa: True } & \\ \text { Tb: False } & \text { Sb: True } & \text { Wab: True } \\ \text { Ra: True } & \text { Va: False } & \text { Wba: True } \\\text { Rb: False } & \text { Vb:True } & \text { Wbb: True }\end{array}$

Question 70

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.

-1. (?x)[Lx ? (?y)(My • Nxy)]
2) (?x)[Lx • (?y)Nxy] / (?x)[Mx • (?y)Nyx]

A) Valid
B) Invalid. Counterexample in a domain of one member, in which:
La: True Ma: True Naa: False
C) Invalid. Counterexample in a domain of two members, in which:
$\begin{array}{llll}\text { La: False } & \text { Ma: True } & \text { Naa: True } & \text { Nba: True } \\\text { Lb:True } & \text { Mb: True } & \text { Nab: True } & \text { Nbb: False }\end{array}$

D) Invalid. Counterexample in a domain of two members, in which:
$\begin{array}{llll}\text { La: True } & \text { Ma: False } & \text { Naa: True } & \text { Nba: False } \\\text { Lb: False } & \text { Mb: True } & \text { Nab: True } & \text { Nbb: False }\end{array}$

E) Invalid. Counterexample in a domain of two members, in which: La: True Ma: False Naa: True Nba: True
Lb: False Mb: True Nab: False Nbb: False

Question 71

select the best translation into predicate logic, using the following translation key:
d: Diego
s: Sean
Dx: x is on the Dean's list
Px: x is a philosophy major
Sx: x is a student
Vx: x is a valedictorian
-There is exactly one philosophy major on the Dean's list.

A) (∃x){(Px • Dx) • (∀y)[(Py • Dy) ⊃ y=x]}
B) (Pd • Ps) ⊃ [(Dd $\lor$ Ds) • ~(Dd • Ds)]
C) (∀x)(∀y){[(Px • Dx) • (Py • Dy)] ⊃ x=y}
D) (∃x){(Px • Dx) ⊃ (∀y)[(Py • Dy) ⊃ x=y]}
E) (Pd • Dd) $\lor$ (Ps • Ds)

Question 72

select the best translation into predicate logic, using the following translation key:
d: Diego
s: Sean
Dx: x is on the Dean's list
Px: x is a philosophy major
Sx: x is a student
Vx: x is a valedictorian
-There are exactly two philosophy majors on the Dean's list.

A) Pd • Ps • Dd • Ds
B) (∃x)(∀y)[(Px • Py • Dx • Dy) ⊃ x=y]
C) (∀x)(∀y)(∀z)[(Px • Py • Pz • Dx • Dy • Dz) ⊃ (x =y $\lor$ x=z $\lor$ y=z)]
D) (∃x)(∃y){Px • Dx • Py • Dy • x≠y • (∀z)[(Pz • Dz) ⊃ (z=x $\lor$ z=y)]}
E) ~(∃x)(∃y)(∃z)(Px • Py • Pz • Dx • Dy • Dz • x≠y • x≠z • y≠z)

Question 73

select the best translation into predicate logic, using the following translation key:
d: Diego
s: Sean
Dx: x is on the Dean's list
Px: x is a philosophy major
Sx: x is a student
Vx: x is a valedictorian
-At least four philosophy majors are on the Dean's list.

A) (∃x)(∃y)(∃z)(∃w)[Px • Dx • Py • Dy • Pz • Dz • Pw • Dw •
(x=y $\lor$ x=z $\lor$ x=w $\lor$ y=z $\lor$ y=w $\lor$ z=w)]
B) (∀x)(∀y)(∀z)(∀w)[(Px • Py • Pz • Pw • Dx • Dy • Dz • Dw) ⊃
(x=y $\lor$ x=z $\lor$ x=w $\lor$ y=z $\lor$ y=w $\lor$ z=w)]
C) (∃x)(∃y)(∃z)(∃w)(Px • Dx • Py • Dy • Pz • Dz • Pw • Dw • x≠y • x≠z • x≠w • y≠z • y≠w • z≠w)
D) (∀x)(∀y)(∀z)(∀w)[(Px • Py • Pz • Pw • Dx • Dy • Dz • Dw) ⊃
(x≠y • x≠z • x≠w • y≠z • y≠w • z≠w)]
E) (∃x)(∃y)(∃z)(∃w)[(Px • Dx • Py • Dy • Pz • Dz • Pw • Dw) ⊃
(x≠y • x≠z • x≠w • y≠z • y≠w • z≠w)]

Question 74

select the best translation into predicate logic, using the following translation key:
d: Diego
s: Sean
Dx: x is on the Dean's list
Px: x is a philosophy major
Sx: x is a student
Vx: x is a valedictorian
-All philosophy majors except Sean are on the Dean's list.

A) (∃x)(Px • Dx • x≠s)
B) Ps • ∼Ds • (∀x)[(Px • x≠s) ⊃ Dx]
C) (∀x)[Dx ⊃ (Px ⊃ x≠s)]
D) (∀x)(Px ⊃ Dx) • Ps • ~Ds
E) (∀x)[Px ⊃ (Dx ⊃ x≠s)]

Question 75

select the best translation into predicate logic, using the following translation key:
d: Diego
s: Sean
Dx: x is on the Dean's list
Px: x is a philosophy major
Sx: x is a student
Vx: x is a valedictorian
-Of the philosophy majors, only Diego is a valedictorian.

A) Vd • (∀x)[Px ⊃ (~Vx $\lor$ x=d)]
B) Pd • Vd • (∀x)[(Px • Vx) ⊃ x=d]
C) ~(∃x)(Px • Vx • x≠d)
D) (∀x)[(Px • Vx) ⊃ x=d]
E) (∃x)(∀y)[x=d • Vd • Pd • (Py ⊃ y=d)]

Question 76

select the best translation into predicate logic, using the following translation key:
d: Diego
s: Sean
Dx: x is on the Dean's list
Px: x is a philosophy major
Sx: x is a student
Vx: x is a valedictorian
-At most two philosophy majors are on the Dean's list.

A) (∃x)(∃y){Px • Dx • Py • Dy • (∀z)[(Pz • Dz) ⊃ (x=z $\lor$ y=z)]}
B) (∃x)(∃y)[Px • Dx • Py • Dy • ~(∃z)(Pz • Dz • x=z • y=z)]
C) (∀x)(∀y)(∀z)(Px • Dx • Py • Dy • Pz • Dz • x≠y • x≠z • y≠z)
D) ~(∃x)(∃y)(∃z)(Px • Dx • Py • Dy • Pz • Dz • x≠y • x≠z • y≠z)
E) (∀x)(∀y)(∀z)[(Px • Dx • Py • Dy • Pz • Dz) ⊃ (x=y $\lor$ x=z $\lor$ y=z)]

Question 77

select the best translation into predicate logic, using the following translation key:
d: Diego
s: Sean
Dx: x is on the Dean's list
Px: x is a philosophy major
Sx: x is a student
Vx: x is a valedictorian
-Exactly one student is a valedictorian.

A) (∃x)(Sx • Vx) • ~(∃y)(Sy • Vy)
B) (∀x)[(Sx • Vx) ⊃ ~(∃y)(Sy • Vy • x≠y)]
C) ~(∃x)(∃y)(Sx • Vx • Sy • Vy • x≠y)
D) (∃x)[Sx • Vx • (∃y)(Sy • Vy • x=y)]
E) (∃x){Sx • Vx • (∀y)[(Sy • Vy) ⊃ y=x]}

Question 78

select the best translation into predicate logic, using the following translation key:
d: Diego
s: Sean
Dx: x is on the Dean's list
Px: x is a philosophy major
Sx: x is a student
Vx: x is a valedictorian
-The valedictorian is Diego.

A) (∃x){Vx • (∀y)[(Vy ⊃ y=x) • x=d]}
B) (∀x)(Vx ⊃ x=d)
C) (∀x)(Vx ≡ x=d)
D) Vd • ~(∃x)(Vx • x=d)
E) (∃x)[Vx • (∃y)(Vy • x=y) • x=d]

Question 79

select the best translation into predicate logic, using the following translation key:
d: Diego
s: Sean
Dx: x is on the Dean's list
Px: x is a philosophy major
Sx: x is a student
Vx: x is a valedictorian
-Only Sean and Diego are both on the Dean's list and philosophy majors.

A) (∀x)[(Dx • Px) ⊃ (x=s • x=d)]
B) (∀x)[(Dx • Px) ⊃ ~(x=s • x=d)]
C) Ds • Ps • Dd • Pd • [(∃x)(Px • Dx) ⊃ ~(x=s • x=d)]
D) Ds • Dd • Ps • Pd • (∀x)[(Dx • Px) ⊃ (x=s $\lor$ x=d)]
E) Ds • Ps • Dd • Pd • (∀x)[(Dx • Px) ⊃ ~(x=s • x=d)]

Question 80

select the best translation into predicate logic, using the following translation key:
d: Diego
s: Sean
Dx: x is on the Dean's list
Px: x is a philosophy major
Sx: x is a student
Vx: x is a valedictorian
-The philosophy major on the Dean's list is a valedictorian.

A) (Ps • Ds • Vs) $\lor$ (Pd • Dd • Vd)
B) (∀x)[Vx ⊃ (Px • Dx)]
C) (∃x){Px • Dx • (∀y)[(Py • Dy) ⊃ y=x] • Vx}
D) (∃x){Px • Vx • (∀y)[(Py • Vy) ⊃ y=x] • Dx}
E) (∃x){Dx • Vx • (∀y)[(Dy • Vy) ⊃ y=x] • Px}