Question 1

use the given interpretations to translate each sentence of predicate logic into natural, English sentences.
f: Fifi 
g: Gigi
Px: x is a poodle 
Qx: x is abused 
Rx: x is loved
Sx: x will fetch balls 
Tx: x will fetch sticks.
-(∀x){(Px • Qx) ⊃ [(Rx ⊃ (∼Sx ⊃ Tx)]}

Accepted Answer

The answer of use the given interpretations to translate each...

Question 2

1. ∼(∃x)Ax
2. (∀x)∼Ax ⊃ ∼(∃x)Bx
-Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

Accepted Answer

A)  ∼(∀x)Ax ⊃ (∃x)∼Bx 
B)  Ax 
C)  ∼Ax 
D)  ∼(∃x)Ax ⊃ (∃x)∼Bx 
E)  ∼(∃x)Ax ⊃ ∼(∃x)Bx

Question 3

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
-1. (∀x)(Fx ⊃ Gx)
2) (∃x)Fx / (∀x)(∼Gx ⊃ ∼Ex)

Accepted Answer

A)  Valid 
B)  Invalid. Counterexample in a domain of 2 members, in which:
Fa: False Ga: False Ea: False
Fb: True Gb: False Eb: True 
C)  Invalid. Counterexample in a domain of 2 members, in which:
Fa: False Ga: False Ea: True
Fb: True Gb: True Eb: True 
D)  Invalid. Counterexample in a domain of 2 members, in which:
Fa: False Ga: False Ea: False
Fb: False Gb: True Eb: False 
E)  Invalid. Counterexample in a domain of 2 members, in which:
Fa: True Ga: False Ea: False
Fb: True Gb: True Eb: True

Question 4

refer to the following formula:
∼(∀x){(Ix • Jx) ⊃ [Kx ≡ (La • Lb)]}
-Which is the main operator of the formula?

Accepted Answer

A)  ∀x 
B)  ≡ 
C)  ⊃ 
D)  • 
E)  ∼

Question 5

select the best translation into predicate logic.
-Some rationalists who are skeptics are not theists.

Accepted Answer

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

Question 6

refer to the following formula: (∀x)[(Ex $ \lor $ Fx) ⊃ (Gx • Hd)]
-List all of the subformulas in the scope of '(∀x)'.

Accepted Answer

The answer of refer to the following formula: (∀x)[(Ex \(...

Question 7

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
-1. (∃x)(Px • Qx)
2) (∃x)(Px • Rx)
3) (∀x)(Qx ⊃ Sx)
4) (∀x)(Rx ⊃ Tx) / (∃x)(Px • Sx) • (∃x)(Px • Tx)

Accepted Answer

A)  Valid 
B)  Invalid. Counterexample in a domain of 1 member, in which:
Pa: True Qa: True Ra: True Sa: True Ta: True 
C)  Invalid. Counterexample in a domain of 1 member, in which:
Pa: False Qa: True Ra: True Sa: True Ta: True 
D)  Invalid. Counterexample in a domain of 1 member, in which:
Pa: True Qa: True Ra: False Sa: True Ta: True 
E)  Invalid. Counterexample in a domain of 1 member, in which:
Pa: True Qa: True Ra: True Sa: False Ta: True

Question 8

select the best translation into predicate logic.
-Some athletes who don't work hard receive scholarships, if, and only if, no athletes who play professional sports don't have determination.

Accepted Answer

A)  (∃x)[(Ax • ∼Wx) • Sx] ≡ (∀x)[(Ax • Px) ⊃ ∼∼Dx] 
B)  (∃x)[(Ax • ∼Wx) • Sx] ⊃ (∀x)[(Ax • Px) ⊃ ∼∼Dx] 
C)  (∃x)[(Ax • ∼Wx) • Sx] ⊃ (∀x)[(Ax • Px) ⊃ ∼Dx] 
D)  (∃x)[(Ax • ∼Wx) • Sx] ≡ (∀x)[(Ax • Px) ⊃ ∼Dx]

Question 9

construct a model for each of the given theories in the following domain by assigning members of the domain to the constants used in the theory and sets of members of the domain to the predicates used in the theory.
Domain = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}
-Aa • Ab
∼Ba • ∼Bb 
(∃x)(Ax • Bx)

Accepted Answer

The answer of construct a model for each of the...

Question 10

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates.
-Some grass is high and thick.

Accepted Answer

A)  (∃x)[Gx $ \lor $ (Hx • Tx)] 
B)  (∃x)[Gx $ \lor $ (Hx $ \lor $ Tx)] 
C)  (∃x)[Gx • (Hx • Tx)] 
D)  (∃x)[Gx ⊃ (Hx $ \lor $ Tx)] 
E)  (∃x)[Gx • (Hx $ \lor $ Tx)]

Question 11

1. (∃x)(Mx • Ox) ⊃ (∃x)Nx
2. (∃x)(Px • Mx)
3. (∀x)(∼Px $ \lor $ Ox)
-Which of the following propositions is derivable from the given premises in M?

Accepted Answer

A)  (∃x)∼Nx 
B)  (∀x)Ox 
C)  (∀x)Nx 
D)  (∃x)Nx 
E)  (∃x)∼Mx

Question 12

Things are pleasant if, and only if, they are not too crowded. Everything too crowded is noisy. So if something isn't noisy, then something is pleasant.
-Which of the following propositions is a good assumption for conditional proof for the above argument?

Accepted Answer

A)  (∃x)∼Nx 
B)  ∼(∃x)Nx 
C)  ∼(∃x)∼Nx 
D)  ∼Nx 
E)  Nx

Question 13

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

Accepted Answer

A)  (∃x)∼Ix 
B)  (∃x)∼Gx 
C)  (∀x)∼Ix 
D)  (∀x)(Gx ⊃ Hx) 
E)  (∀x)(Gx • Ix)

Question 14

select the best translation into predicate logic.
-All cities are productive if and only if they are both nicely placed and not on the water.

Accepted Answer

A)  (∀x){Cx ⊃ [Px ≡ (Nx • ∼Wx)]} 
B)  (∀x){Px ⊃ [Cx ≡ (Nx • ∼Wx)]} 
C)  (∀x){Px ⊃ [Cx ≡ ∼ (Nx • Wx)]} 
D)  (∀x){Cx ⊃ [Px ≡ ∼ (Nx • Wx)]}

Question 15

(∃x)(Px • Qx) ⊃ [(∃x)Px • (∃x)Qx]
-Consider assuming '(∃x)(Px • Qx)' for a conditional proof of the above logical truth. Which
Of the following propositions is a legitimate second step in that proof?

Accepted Answer

A)  (Px • Qx) ⊃ [(∃x)Px • (∃x)Qx] 
B)  Px • Qx 
C)  (∃x)Px 
D)  Pb • Qb 
E)  (∃x)Qx

Question 16

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)  (∃x)[Px • (Ix • ∼Ox)] (∃x)[Px • (Ox • ∼Ix)] (∀x)[Px ⊃ (Ix ≡ ∼Ox)] 
B)  (∃x)[Px • (Ix • ∼Ox)] (∃x)[Px • (Ox • ∼Ix)] (∀x)[Px ⊃ (Ix ≡ Ox)] 
C)  (∃x)[Px • (Ix • Ox)] (∃x)[Px • (Ox • ∼Ix)] (∀x)[Px ⊃ (Ix ≡ ∼Ox)] 
D)  (∃x)[Px • (Ix • ∼Ox)] (∃x)[Px • (∼Ox • ∼Ix)]
(∀x)[Px ⊃ (Ix ≡ ∼Ox)] 
E)  (∃x)[Px • (Ix • ∼Ox)] (∃x)[Px • (Ox • ∼Ix)] (∀x)[Ix ⊃ (Px ≡ ∼Ox)]

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)(Cx ⊃ Dx)
2. (∀x)(Ex ⊃ ∼Dx) / (∀x)(Ex ⊃ ∼Cx)

Accepted Answer

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

Question 18

Some materialist empiricists are either libertarians or hard determinists. But no empiricist is a hard determinist. So some materialists are libertarians.
-Which of the following claims can also be derived from the premises of this argument?

Accepted Answer

A)  (∃x)[(Mx • Ex) • (Lx • ∼Dx)] 
B)  (∀x)∼Dx 
C)  (∀x)Ex 
D)  (∃x) ∼Lx 
E)  (∀x)∼Lx

Question 19

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates.
-No visitor stayed for dinner.

Accepted Answer

A)  ∼(∀x)(Vx ⊃ Sx) 
B)  ∼(∀x)(Vx ⊃ ∼Sx) 
C)  (∀x) (Vx ⊃ Sx) 
D)  (∀x)∼(Vx ⊃ ∼Sx) 
E)  (∀x)(Vx ⊃ ∼Sx)

Question 20

1. (∀x)(Jx ⊃ Kx)
2. (∀x)(Jx ⊃ ∼Lx)
-Consider assuming 'Jx' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in M of the given premises with that further assumption for conditional proof?

Accepted Answer

A)  Kx 
B)  ∼Lx 
C)  (∀x)Jx 
D)  Jx ⊃ Kx 
E)  Ja ⊃ ∼Lx

Exam 4: Monadic Predicate Logic

use the given interpretations to translate each sentence of predicate logic into natural, English sentences. f: Fifi g: Gigi Px: x is a poodle Qx: x is abused Rx: x is loved Sx: x will fetch balls Tx: x will fetch sticks. -(∀x){(Px • Qx) ⊃ [(Rx ⊃ (∼Sx ⊃ Tx)]}

1. ∼(∃x) Ax 2. (∀x) ∼Ax ⊃ ∼(∃x) Bx -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample. -1. (∀x) (Fx ⊃ Gx) 2) (∃x) Fx / (∀x) (∼Gx ⊃ ∼Ex)

refer to the following formula: ∼(∀x) {(Ix • Jx) ⊃ [Kx ≡ (La • Lb) ]} -Which is the main operator of the formula?

select the best translation into predicate logic. -Some rationalists who are skeptics are not theists.

refer to the following formula: (∀x)[(Ex $\lor$ Fx) ⊃ (Gx • Hd)] -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) (Px • Qx) 2) (∃x) (Px • Rx) 3) (∀x) (Qx ⊃ Sx) 4) (∀x) (Rx ⊃ Tx) / (∃x) (Px • Sx) • (∃x) (Px • Tx)

select the best translation into predicate logic. -Some athletes who don't work hard receive scholarships, if, and only if, no athletes who play professional sports don't have determination.

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates. -Some grass is high and thick.

1. (∃x) (Mx • Ox) ⊃ (∃x) Nx 2. (∃x) (Px • Mx) 3. (∀x) (∼Px $\lor$ Ox) -Which of the following propositions is derivable from the given premises in M?

Things are pleasant if, and only if, they are not too crowded. Everything too crowded is noisy. So if something isn't noisy, then something is pleasant. -Which of the following propositions is a good assumption for conditional proof for the above argument?

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

select the best translation into predicate logic. -All cities are productive if and only if they are both nicely placed and not on the water.

(∃x) (Px • Qx) ⊃ [(∃x) Px • (∃x) Qx] -Consider assuming '(∃x) (Px • Qx) ' for a conditional proof of the above logical truth. Which Of the following propositions is a legitimate second step in that proof?

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)(Cx ⊃ Dx) 2. (∀x)(Ex ⊃ ∼Dx) / (∀x)(Ex ⊃ ∼Cx)

Some materialist empiricists are either libertarians or hard determinists. But no empiricist is a hard determinist. So some materialists are libertarians. -Which of the following claims can also be derived from the premises of this argument?

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates. -No visitor stayed for dinner.

1. (∀x) (Jx ⊃ Kx) 2. (∀x) (Jx ⊃ ∼Lx) -Consider assuming 'Jx' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in M of the given premises with that further assumption for conditional proof?