Question 1

select the best translation into predicate logic, using the following translation key: 
a: one
b: two 
c: three
f(x): the successor of x 
g(x,y): the sum of x and y 
Nx: x is a number
Dxy: x is divisible by y 
Gxy: x is greater than y
-If a number is divisible by three, then its successor is not.

Accepted Answer

A)   (∀x) {Nx ⊃ [Dxc ⊃ ∼Df(x) c]}
B)   (∀x) {Nx ⊃ [Dxc ⊃ Df(x) c]}
C)   (∀x) {Nx ⊃ [∼Dxc ⊃ Df(x) c]}
D)   (∀x) {Nx ⊃ [Dxc ⊃ Df(x) ∼c]}
E)   (∀x) [Nx ⊃ Df(x) c] A

Question 2

1. (∀x)[(Hx • x≠a) ⊃ Jax]
2. Hb • Ib • Mb • (∀x)[(Ix • Mx) ⊃ x=b]
3. Ha • ∼Ma
-Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

Accepted Answer

A)   ~Ma • Mb
B)   (∀x) [(Ix • Mx)  ⊃ x=b] • Mb
C)   (Hb • b≠a)  ⊃ Jab
D)   Hb • Ib • Mb • [(Ia • Ma)  ⊃ a=b]
E)   (Ha • a≠a)  ⊃ Jax C

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.

Accepted Answer

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

Question 4

select the best translation into predicate logic, using the following translation key: 
a: one
b: two 
c: three
f(x): the successor of x 
g(x,y): the sum of x and y 
Nx: x is a number
Dxy: x is divisible by y 
Gxy: x is greater than y
-Any number is divisible by two if, and only if, the successor of its successor is divisible by two.

Accepted Answer

A)   (∀x) [Nx ⊃ Df(f(b) ) c]
B)   (∀x) {Nx ≡ [Dxb • Df(f(x) ) b]}
C)   (∀x) {Nx ⊃ [Dbx ≡ Df(x) b]}
D)   (∀x) {(Nx ⊃ Dxb)  ≡ Df(f(x) ) b}
E)   (∀x) {Nx ⊃ [Dxb ≡ Df(f(x) ) b}

Question 5

derive the conclusions of each of the following arguments using the rules of inference for F.
-1. (∀x)[Ax ⊃ (∃y)(By • Cxy)]
2. (∃x)(Ax • Dx)
3. (∀x)(Bx ⊃ Ex) / (∃x)[Dx • (∃y)(Ey • Cxy)]

Accepted Answer

The answer of derive the conclusions of each of the...

Question 6

Translate each sentence into predicate logic, using the given translation keys.
use:
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.

Accepted Answer

The answer of Translate each sentence into predicate logic, using...

Question 7

1. (∃x)(Ax • Bx • Cax)
2. (∀x)[(Ax • Cax) ⊃ x=d]
-Which of the following propositions is derivable from the given premises in F?

Accepted Answer

A)   Bd
B)   Caa
C)   Ax
D)   a=d
E)   Aa

Question 8

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.

Accepted Answer

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 9

1. (∃x)(∃y)(Dx • Exb • Dy • Eyb • x≠y)
2. (∀x)(Dx ⊃ Ebx)
-Which of the following propositions is derivable from the given premises in F?

Accepted Answer

A)   ~(∃x) (Dx • Exx) 
B)   (∃x) (∃y) (Ebx • Exb • Eby • Eyb • x≠y) 
C)   (∀x) (Ebx ⊃ Dx) 
D)   ~(∃x) (∃y) (Dx • Dy • Exy) 
E)   (∀x) (∀y) [x≠y ⊃ (Dx  ∨ \lor ∨  Dy) ]

Question 10

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.

Accepted Answer

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 11

derive the conclusions of each of the following arguments using the rules of inference for F.
-1. (∃x)[Dx • (∀y)(Ey ⊃ Fxy)]
2. (∀x)(Dx ⊃ Ex) / (∃y)(Ey • Fyy)

Accepted Answer

The answer of derive the conclusions of each of the...

Question 12

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

Accepted Answer

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 13

1. Fab • (∀x)(Fax ⊃ x=b)
2. ∼Fac
-Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

Accepted Answer

A)   Faa ⊃ a=b
B)   Fab
C)   Fax ⊃ x=b) 
D)   Fab ⊃ b=b
E)   Fac

Question 14

construct a derivation of each logical truth of
-(∀x)(∀y)(Pxy ≡ ∼Pyx) ⊃ [(∃x)(∃y)Pxy ⊃ ∼(∀x)(∀y)Pxy]

Accepted Answer

The answer of construct a derivation of each logical truth...

Question 15

use:
b: Bhavin 
c: Chloe 
m: Megha
n: Nietzsche 
p: Plato
Ax: x is an altruist 
Jx: x is joyful
Px: x is a philosopher 
Rx: x is Russian
Tx: x is thoughtful 
Bxy: x is a brother of y 
Mxy: x mocks y
Rxy: x is richer than y
Sxy: x is smarter than y
-The richest philosopher is smarter than any of Chloe's brothers.

Accepted Answer

The answer of use:
b: Bhavin 
c: Chloe 
m: Megha
n: Nietzsche...

Question 16

determine whether the given argument is valid or invalid. If it is valid, derive the conclusion using our rules of inference and equivalence. If it is invalid, provide a counterexample.
-1. (?x)[Lx ? (?y)(My • Nxy)]
2. (?x)[Lx • (?y)Nxy] / (?x)[Mx • (?y)Nxy]

Accepted Answer

The answer of determine whether the given argument is valid...

Question 17

select the best translation into predicate logic, using the following translation key:
b: Britain
c: Charles 
e: Elizabeth 
t: Tescos
Px: x is a person
Qx: x is a Queen of England
Wx: x is a woman
Exy: x is more exalted than y 
Ixy: x is in y
Pxy: x shops at y 
Sxy: x is a son of y
-There is at most one queen in Britain.

Accepted Answer

A)   Qe • ~(∃x) (Qx • x≠e) 
B)   ~(∃x) (∃y) (Qx • Ixb • Qy • Iyb) 
C)   (∀x) [(Qx • Ixb)  ⊃ ~(∃y) (Qy • Iyb • y=x) ]
D)   (∀x) (∀y) [(Qx • Qy)  ⊃ (~Ixb  ∨ \lor ∨  ~Iyb) ]
E)   (∀x) (∀y) [(Qx • Ixb • Qy • Iyb)  ⊃ x=y]

Question 18

1. (∀x)(∀y)[(Px • Py) ⊃ Pf(x,y)]
2. (∃x)[Px • (∃y)(Py • x≠y • Rxy)]
-Which of the following propositions is an immediate (one-step) consequence in FF of the given premises?

Accepted Answer

A)   Pf(a)  • (∃y) [Py • f(a) ≠x • Rf(a) y]
B)   Pa • (∃y) (Py • a≠y • Ray) 
C)   Pf(a) 
D)   (∀y) [(Pa • Py)  ⊃ Pf(a) ]
E)   (∀y) [(Pf(a)  • Py)  ⊃ Pf(a,y) ]

Question 19

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?

Accepted Answer

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 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
-No lion is faster than some tigers.

Accepted Answer

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) ]

Exam 5: Full First-Order Logic

select the best translation into predicate logic, using the following translation key: a: one b: two c: three f(x) : the successor of x g(x,y) : the sum of x and y Nx: x is a number Dxy: x is divisible by y Gxy: x is greater than y -If a number is divisible by three, then its successor is not.

1. (∀x) [(Hx • x≠a) ⊃ Jax] 2. Hb • Ib • Mb • (∀x) [(Ix • Mx) ⊃ x=b] 3. Ha • ∼Ma -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

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.

derive the conclusions of each of the following arguments using the rules of inference for F. -1. (∀x)[Ax ⊃ (∃y)(By • Cxy)] 2. (∃x)(Ax • Dx) 3. (∀x)(Bx ⊃ Ex) / (∃x)[Dx • (∃y)(Ey • Cxy)]

Translate each sentence into predicate logic, using the given translation keys. use: 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.

1. (∃x) (Ax • Bx • Cax) 2. (∀x) [(Ax • Cax) ⊃ x=d] -Which of the following propositions is derivable from the given premises in F?

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.

1. (∃x) (∃y) (Dx • Exb • Dy • Eyb • x≠y) 2. (∀x) (Dx ⊃ Ebx) -Which of the following propositions is derivable from the given premises in F?

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.

derive the conclusions of each of the following arguments using the rules of inference for F. -1. (∃x)[Dx • (∀y)(Ey ⊃ Fxy)] 2. (∀x)(Dx ⊃ Ex) / (∃y)(Ey • Fyy)

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

1. Fab • (∀x) (Fax ⊃ x=b) 2. ∼Fac -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

construct a derivation of each logical truth of -(∀x)(∀y)(Pxy ≡ ∼Pyx) ⊃ [(∃x)(∃y)Pxy ⊃ ∼(∀x)(∀y)Pxy]

determine whether the given argument is valid or invalid. If it is valid, derive the conclusion using our rules of inference and equivalence. If it is invalid, provide a counterexample. -1. (?x)[Lx ? (?y)(My • Nxy)] 2. (?x)[Lx • (?y)Nxy] / (?x)[Mx • (?y)Nxy]

1. (∀x) (∀y) [(Px • Py) ⊃ Pf(x,y) ] 2. (∃x) [Px • (∃y) (Py • x≠y • Rxy) ] -Which of the following propositions is an immediate (one-step) consequence in FF of the given premises?

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.