Question 1

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates.
-Efraim takes acting classes if, and only if, he gets time off from work.

Accepted Answer

A)  Ex ≡ Wy 
B)  Ae ⊃ We 
C)  ∼Ae $ \lor $ We 
D)  We ⊃ Ae 
E)  Ae ≡ We

Question 2

select the best translation into predicate logic.
-All tall athletes work hard.

Accepted Answer

A)  (∀x)[Wx ⊃ (Tx • Ax)] 
B)  (∀x)[(Tx $ \lor $ Ax) ⊃ Wx] 
C)  (∀x)[Wx ⊃ (Tx $ \lor $ Ax)] 
D)  (∀x)[(Tx • Ax) ⊃ Wx]

Question 3

construct theories for which the following interpretation is a model (i.e. construct at least two sentences which are true under the given interpretation).
Domain = {1, 2, 3, ..., 28, 29, 30}
E = {2, 4, 6, ..., 28, 30}
O = {1, 3, 5, ..., 27, 29}
P = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
a = 1 d = 19
b = 2 e = 23
c = 3 f = 29
-Construct a theory of at least two sentences, at least one of which uses an existential quantifier.

Accepted Answer

The answer of construct theories for which the following interpretation...

Question 4

consider the following domain, assignment of objects in the domain, and assignments sets to predicates.
Domain = {1, 2, 3, ..., 28, 29, 30}
N = {1, 2, 3, ..., 28, 29, 30}
E = {2, 4, 6, ..., 28, 30}
O = {1, 3, 5, ..., 27, 29}
P = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
a = 1
b = 2
c = 28
-Given the customary truth tables, which of the following theories is modeled by the above interpretation?

Accepted Answer

A)  (∃x)[Nx • (Ox • Px)] (∃x)[Nx • (Ox • ∼Px)] (∀x)[Nx ⊃ (Ox • Ex)] 
B)  (∃x)[Nx • (Ox • Px)] (∃x)[Nx • (Ox • ∼Px)] (∀x)[Nx ⊃ (Ox $ \lor $ Ex)] 
C)  (∀x)[Nx ⊃ (Ox $ \lor $ Px)] (∃x)[Nx • (∼Ox • ∼Px)] (∀x)[Nx ⊃ (Ox $ \lor $ Ex)] 
D)  (∀x)[Nx • (Ox $ \lor $ Px)] (∃x)[Nx • (∼Ox • ∼Px)] (∀x)[Nx ⊃ (Ox $ \lor $ Ex)] 
E)  (∃x)[Nx • (Ox • Px)] (∀x)[Nx ⊃ (Ox • ∼Px)] (∀x)[Nx ⊃ (Ox $ \lor $ Ex)]

Question 5

determine whether the given argument is valid or invalid. If it is valid, provide a derivation of the conclusion from the premises. If it is invalid, provide a counterexample.
-1. (∃x)[(Dx • Ex) • Fx]
2. (∀x)[(Dx • Fx) ⊃ ∼Gx] / (∃x)(Ex • ∼Gx)

Accepted Answer

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

Question 6

translate the given paragraphs into arguments written in M, using the given translation key. Then, derive their conclusions using the rules of inference for M.
-Some materialist empiricists are either libertarians or hard determinists. But no empiricist is a hard determinist. So some materialists are libertarians. (Dx: x is a hard determinist; Ex: x is an empiricist; Lx: x is a libertarian; Mx: x is a materialist)

Accepted Answer

The answer of translate the given paragraphs into arguments written...

Question 7

refer to the following formula: ∼(∀x){(Ix • Jx) ⊃ [Kx ≡ (La • Lb)]}
-Are there any free variables? If so, which are they?

Accepted Answer

The answer of refer to the following formula: ∼(∀x){(Ix •...

Question 8

determine whether the given formula is a logical truth of M or not. If it is a logical truth, provide a proof of the formula. If it is not a logical truth, provide a counterexample in a finite domain.
-(∀x)(Ax ⊃ ∼Bx) $ \lor $ (∃x)(Ax • Bx)

Accepted Answer

The answer of determine whether the given formula is a...

Question 9

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} 
P = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}
I = {Mercury, Venus}
O = {Mars, Jupiter, Saturn, Uranus, Neptune}
a = Mercury b = Jupiter c = Saturn
d = Pluto
-Given the customary truth tables, which of the following theories is modeled by the above interpretation?

Accepted Answer

A)  Ia • (Ob • Oc)
(∀x)(Ix $ \lor $ Ox)
∼(∃x)(Ix • Ox) 
B)  Ia • (Ib • Oc)
∼(∀x)(Ix $ \lor $ Ox)
∼(∃x)(Ix • Ox) 
C)  Ia • (Ob • Oc)
(∀x)(Ix $ \lor $ ∼Ox)
∼(∃x)(Ix • Ox) 
D)  Ia • (Ob • Oc)
∼(∀x)(Ix $ \lor $ Ox)
∼(∃x)(Ix • Ox) 
E)  Ia • (Pb • Od)
∼(∃x)(Ix $ \lor $ Ox)
∼(∃x)(Ix • Ox)

Question 10

1. (∃x)(Gx • ∼Hx) ≡ (∃x)(Hx • ∼Ix)
2. (∀x)(Hx ⊃ Ix)
-Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

Accepted Answer

A)  Ga • ∼Ha ≡ (∃x)(Hx • ∼Ix) 
B)  (∃x)(Gx • ∼Hx) ≡ ∼(∀x)∼(Hx • ∼Ix) 
C)  ∼(∀x)(Gx • ∼Hx) ≡ (∃x)(Hx • ∼Ix) 
D)  Hc ⊃ Ia 
E)  (Ga • ∼Ha) ≡ (Ha • ∼Ia)

Question 11

1. ∼(∃x)[Fx • (Gx • Hx)]
2. ∼(∃x)(Ix • ∼Fx)
-Which of the following propositions is derivable from the given premises in M?

Accepted Answer

A)  (∀x)(∼Gx $ \lor $ ∼Hx) 
B)  (∀x)[(Ix $ \lor $ (∼Gx • ∼Hx)] 
C)  (∀x)[Ix ⊃ (∼Gx $ \lor $ ∼Hx)] 
D)  (∃x)(∼Gx $ \lor $ ∼Hx) 
E)  (∃x)Fx

Question 12

select the best translation into predicate logic.
-There are no settlements on the water with trading ports.

Accepted Answer

A)  (∀x)[(Sx • Wx) ⊃ ∼Tx] 
B)  ∼(∀x)[(Sx • Wx) ⊃ Tx] 
C)  (∀x)[(Sx ⊃ ∼(Wx $ \lor $ Tx)] 
D)  (∀x)[(Sx ⊃ (∼Wx • ∼Tx)]

Question 13

select the best translation into predicate logic.
-All cities and settlements are nicely placed and productive.

Accepted Answer

A)  (∀x)[(Cx • Sx) ⊃ (Nx • Px)] 
B)  (∀x)[(Cx • Sx) • (Nx • Px)] 
C)  (∀x)[(Cx $ \lor $ Sx) • (Nx • Px)] 
D)  (∀x)[(Cx $ \lor $ Sx) ⊃ (Nx • Px)]

Question 14

refer to the following formula: (∃x)[Mx • (∼Nc $ \lor $ ∼Ox)] ≡ (Py • Pb)
-List all of the subformulas in the scope of '(∃x)'.

Accepted Answer

The answer of refer to the following formula: (∃x)[Mx •...

Question 15

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
-1. (?x)(Ax ? Bx) ? (?x)Cx
2) (?x)(Ax • ?Bx)
3) (?x)(Dx ? Bx) / (?x)(Dx ? Cx)

Accepted Answer

A)  Valid 
B)  Invalid. Counterexample in a domain of 2 members, in which:
$$\begin{array} { l l l l } 
\text { Aa: True } & \text { Ba: False } & \text { Ca: False } & \text { Da: False } \
\text { Ab: False} &\text { Bb: True } &\text {Cb: False }&\text { Db: True } \
\end{array}$$ 
C)  Invalid. Counterexample in a domain of 2 members, in which:
$$\begin{array} { l l l l } 
\text { Aa: True } & \text { Ba: False } & \text { Ca: False } & \text { Da: False } \
\text { Ab: True} &\text { Bb: True } &\text {Cb: True }&\text { Db: True } \
\end{array}$$ 
D)  Invalid. Counterexample in a domain of 2 members, in which:
$$\begin{array} { l l l l } 
\text { Aa: False} & \text { Ba: False } & \text { Ca: False } & \text { Da: False } \
\text { Ab: False} &\text { Bb: True } &\text {Cb: False }&\text { Db: True } \
\end{array}$$ 
E)  Invalid. Counterexample in a domain of 2 members, in which:
Aa: True Ba: False Ca: False Da: False Ab: True Bb: False Cb: False Db: True

Question 16

refer to the following formula: (∀x)[(Ex $ \lor $ Fx) ⊃ (Gx • Hd)]
-Which is the main operator of the formula?

Accepted Answer

A)  ∀x 
B)  $ \lor $ 
C)  ⊃ 
D)  • 
E)  None of the above.

Question 17

derive the conclusions of each of the following arguments using the rules of inference for M. Do not use conditional or indirect proof.
-1. (∃x)∼Ix ⊃ (∀x)(Jx $ \lor $ Kx)
2. ∼(∀x)Ix • ∼Jb / Kb

Accepted Answer

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

Question 18

refer to the following formula: (∃x)[(Ax • ∼Bx) • ∼(Cx $ \lor $ Dx)]
-Which variables are bound by the '(∃x)'?

Accepted Answer

A)  The x that follows the A. 
B)  The x that follows the B. 
C)  The x that follows the C. 
D)  The x that follows the D. 
E)  All of the above.

Question 19

Translate each sentence into predicate logic, using the given translation keys.
Ax: x is an athlete
Dx: x has determination
Px: x plays professional sports 
Sx: x receives a scholarship 
Tx: x is tall
Wx: x works hard
-Only athletes who are tall and work hard play professional sports.

Accepted Answer

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

Question 20

determine whether the given formula is a logical truth of M or not. If it is a logical truth, provide a proof of the formula. If it is not a logical truth, provide a counterexample in a finite domain.
-(∃x)(Cx • Dx) $ \lor $ (∃x)(Cx • ∼Dx)

Accepted Answer

The answer of determine whether the given formula is a...

Exam 4: Monadic Predicate Logic

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates. -Efraim takes acting classes if, and only if, he gets time off from work.

select the best translation into predicate logic. -All tall athletes work hard.

determine whether the given argument is valid or invalid. If it is valid, provide a derivation of the conclusion from the premises. If it is invalid, provide a counterexample. -1. (∃x)[(Dx • Ex) • Fx] 2. (∀x)[(Dx • Fx) ⊃ ∼Gx] / (∃x)(Ex • ∼Gx)

refer to the following formula: ∼(∀x){(Ix • Jx) ⊃ [Kx ≡ (La • Lb)]} -Are there any free variables? If so, which are they?

determine whether the given formula is a logical truth of M or not. If it is a logical truth, provide a proof of the formula. If it is not a logical truth, provide a counterexample in a finite domain. -(∀x)(Ax ⊃ ∼Bx) $\lor$ (∃x)(Ax • Bx)

1. (∃x) (Gx • ∼Hx) ≡ (∃x) (Hx • ∼Ix) 2. (∀x) (Hx ⊃ Ix) -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

1. ∼(∃x) [Fx • (Gx • Hx) ] 2. ∼(∃x) (Ix • ∼Fx) -Which of the following propositions is derivable from the given premises in M?

select the best translation into predicate logic. -There are no settlements on the water with trading ports.

select the best translation into predicate logic. -All cities and settlements are nicely placed and productive.

refer to the following formula: (∃x)[Mx • (∼Nc $\lor$ ∼Ox)] ≡ (Py • Pb) -List all of the subformulas in the scope of '(∃x)'.

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample. -1. (?x) (Ax ? Bx) ? (?x) Cx 2) (?x) (Ax • ?Bx) 3) (?x) (Dx ? Bx) / (?x) (Dx ? Cx)

refer to the following formula: (∀x) [(Ex $\lor$ Fx) ⊃ (Gx • Hd) ] -Which is the main operator of the formula?

derive the conclusions of each of the following arguments using the rules of inference for M. Do not use conditional or indirect proof. -1. (∃x)∼Ix ⊃ (∀x)(Jx $\lor$ Kx) 2. ∼(∀x)Ix • ∼Jb / Kb

refer to the following formula: (∃x) [(Ax • ∼Bx) • ∼(Cx $\lor$ Dx) ] -Which variables are bound by the '(∃x) '?

Translate each sentence into predicate logic, using the given translation keys. Ax: x is an athlete Dx: x has determination Px: x plays professional sports Sx: x receives a scholarship Tx: x is tall Wx: x works hard -Only athletes who are tall and work hard play professional sports.

determine whether the given formula is a logical truth of M or not. If it is a logical truth, provide a proof of the formula. If it is not a logical truth, provide a counterexample in a finite domain. -(∃x)(Cx • Dx) $\lor$ (∃x)(Cx • ∼Dx)