Question 1

provide a conterexample in a finite domain to each given invalid argument.
-1. (∃x)(Gx • Hx)
2. (∃x)(Gx • Jx)
3. (∀x)(Jx ⊃ Kx) / (∃x)(Hx • Jx)

Accepted Answer

Counterexample in 2-member domain in which:
 $\begin{array} { l  }
 \text {Ga: True}& \text {Ha: True}& \text { Ja:False}& \text {Ka: True}\
 \text {Gb:True }& \text {Hb: False}& \text { Jb:True}& \text {Kb: True}\
\end{array}$

Question 2

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates.
-Abhishek loves ice cream and pizza.

Accepted Answer

A)  Ai • ∼Ap 
B)  aI • aP 
C)  Ai ⊃ Ap 
D)  Ia • Pa 
E)  Ia ⊃ Pa 
A)  Ai • ∼Ap 
B)  aI • aP 
C)  Ai ⊃ Ap 
D)  Ia • Pa 
E)  Ia ⊃ Pa D

Question 3

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)(Ax • Bx) ⊃ (∀x) Dx
2. ∼Da / (∀x)(Ax ⊃ ∼Bx)

Accepted Answer

$\begin{array}{ll}
\text { 1. }(\exists x)(A x \cdot B x) \supset(\forall x) D x\
\text { 2. } \sim \mathrm{Da} & /(\forall \mathrm{x})(\mathrm{Ax} \supset \sim \mathrm{Bx}) \
\text { 3. }(\exists \mathrm{x}) \sim \mathrm{Dx} & \text { 2, EG } \
\text { 4. } \sim(\forall \mathrm{x}) \mathrm{Dx} & \text { 3, QE } \
\text { 5. } \sim(\exists \mathrm{xx})(\mathrm{Ax} \cdot \mathrm{Bx}) & 1,4, \mathrm{MT} \
\text { 6. }(\forall \mathrm{x}) \sim(\mathrm{Ax} \cdot \mathrm{Bx}) & \text { 5, QE } \
\text { 7. }(\forall \mathrm{x})(\sim \mathrm{Ax} \vee \sim \mathrm{Bx}) & 6, \mathrm{DM} \
\text { 8. }(\forall \mathrm{x})(\mathrm{Ax} \supset \sim \mathrm{Bx}) & \text { 7, Impl }
\end{array}$

Question 4

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)(Ax ⊃ Bx) ⊃ (∃x)Cx
2. (∃x)(Ax • ∼Bx)
3. (∀x)(Dx ⊃ Bx) / (∀x)(Dx ⊃ Cx)

Accepted Answer

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

Question 5

use: t: Tortuga
Bx: x creates bricks 
Cx: x is a city
Nx: x is nicely placed 
Px: x is productive 
Sx: x is a settlement
Tx: x has a trading port 
Wx: x is on the water
-Either Tortuga is a city or it is not a settlement.

Accepted Answer

The answer of use: t: Tortuga
Bx: x creates bricks 
Cx:...

Question 6

Translate each of the following sentences into predicate logic, using the given constants and predicates.
-Farzona's dropping art history is a sufficient condition for her being unhappy. (f: Farzona; Ax: x drops art history; Ux: x is unhappy)

Accepted Answer

The answer of Translate each of the following sentences into...

Question 7

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates.
-Some blankets are not soft.

Accepted Answer

A)  (∃x)(Bx • ∼Sx) 
B)  ∼ (∃x)(Bx • Sx) 
C)  ∼ (∃x)(Bx • ∼Sx) 
D)  ∼ (∃x)( ∼Bx • ∼Sx) 
E)  ∼ (∃x)(Bx ⊃ ∼Sx) 
A)  (∃x)(Bx • ∼Sx) 
B)  ∼ (∃x)(Bx • Sx) 
C)  ∼ (∃x)(Bx • ∼Sx) 
D)  ∼ (∃x)( ∼Bx • ∼Sx) 
E)  ∼ (∃x)(Bx ⊃ ∼Sx)

Question 8

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates.
-Some cherries are red.

Accepted Answer

A)  Sc • Rc 
B)  (∃x)(Cx ⊃ Rx) 
C)  Cs • Cr 
D)  (∃x)(Cx • Rx) 
E)  (∀x)(Cx • Rx) 
A)  Sc • Rc 
B)  (∃x)(Cx ⊃ Rx) 
C)  Cs • Cr 
D)  (∃x)(Cx • Rx) 
E)  (∀x)(Cx • Rx)

Question 9

select the best translation into predicate logic.
-Either Tortuga is a city or it is not a settlement.

Accepted Answer

A)  Ct $ \lor $ ∼St 
B)  tC $ \lor $ ∼tS 
C)  (∃x)(Cx $ \lor $ ∼Sx) 
D)  (∃x)(Cx • ∼Sx) 
A)  Ct $ \lor $ ∼St 
B)  tC $ \lor $ ∼tS 
C)  (∃x)(Cx $ \lor $ ∼Sx) 
D)  (∃x)(Cx • ∼Sx)

Question 10

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.
-(Pf • Pg) • [(Rf • Rg) • (Sf • ∼Sg)]

Accepted Answer

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

Question 11

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

Question 12

1. ∼(∃x)Ax
2. (∀x)∼Ax ⊃ ∼(∃x)Bx
-Which of the following propositions is derivable from the given premises in M?

Accepted Answer

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

Question 13

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)
(∃x)∼Px 
B)  (∃x)(Px • ∼Ix) (∃x)∼Px 
C)  (∃x)(Px • Ix)
(∀x)(Px ⊃ Ix) 
D)  (∃x)(Ox • Ix)
(∃x)∼Px 
E)  (∃x)(Ox • ∼Px) (∀x)(Ix ⊃ Px) 
A)  (∃x)(∼Px • Ix)
(∃x)∼Px 
B)  (∃x)(Px • ∼Ix) (∃x)∼Px 
C)  (∃x)(Px • Ix)
(∀x)(Px ⊃ Ix) 
D)  (∃x)(Ox • Ix)
(∃x)∼Px 
E)  (∃x)(Ox • ∼Px) (∀x)(Ix ⊃ Px)

Question 14

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

Accepted Answer

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

Question 15

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates.
-If Carla works for an airline, then Darlene doesn't.

Accepted Answer

A)  Ad $ \lor $ ∼Ac 
B)  Ax ⊃ ∼Ay 
C)  Ac ≡ ∼Ad 
D)  Ac • ∼Ad 
E)  Ac ⊃ ∼Ad 
A)  Ad $ \lor $ ∼Ac 
B)  Ax ⊃ ∼Ay 
C)  Ac ≡ ∼Ad 
D)  Ac • ∼Ad 
E)  Ac ⊃ ∼Ad

Question 16

1. (∀x)[Ax ⊃ (Bx ⊃ Cx)]
2. ∼(∀x)(Bx ⊃ Dx)
-Which of the following propositions is derivable in M from the given premises?

Accepted Answer

A)  (∀x)(Ax ⊃ Dx) 
B)  (∀x)Ax ⊃ (∃x)(Cx • ∼Dx) 
C)  (∀x)Ax ⊃ (∀x)Dx 
D)  (∀x)[Ax ⊃ (Cx • ∼Dx)] 
E)  (∀x)Ax ⊃ (∀x)(Cx • ∼Dx) 
A)  (∀x)(Ax ⊃ Dx) 
B)  (∀x)Ax ⊃ (∃x)(Cx • ∼Dx) 
C)  (∀x)Ax ⊃ (∀x)Dx 
D)  (∀x)[Ax ⊃ (Cx • ∼Dx)] 
E)  (∀x)Ax ⊃ (∀x)(Cx • ∼Dx)

Question 17

select the best English interpretation of the given statements of predicate logic. 
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]

Accepted Answer

A)  If poodles are loved, then they are not abused. 
B)  Only abused poodles are loved. 
C)  All abused poodles are loved. 
D)  No abused poodles are loved. 
A)  If poodles are loved, then they are not abused. 
B)  Only abused poodles are loved. 
C)  All abused poodles are loved. 
D)  No abused poodles are loved.

Question 18

Translate each of the following sentences into predicate logic, using the given constants and predicates.
-Some excellent doctors are not gentle. (Dx: x is a doctor; Ex: x is excellent; Gx: x is gentle)

Accepted Answer

The answer of Translate each of the following sentences into...

Question 19

Some websites have open comments which are not anonymous. Any website is either anonymous or requires a login. So something with open comments requires a login.
-Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

Accepted Answer

A)  Wn ⊃ (Ax $ \lor $ Lx) 
B)  Wx • (Ox • ∼Ax) 
C)  Wx ⊃ (Ax $ \lor $ Lx) 
D)  Wb • (Oc • ∼Ac) 
E)  Wx 
A)  Wn ⊃ (Ax $ \lor $ Lx) 
B)  Wx • (Ox • ∼Ax) 
C)  Wx ⊃ (Ax $ \lor $ Lx) 
D)  Wb • (Oc • ∼Ac) 
E)  Wx

Question 20

use: t: Tortuga
Bx: x creates bricks 
Cx: x is a city
Nx: x is nicely placed 
Px: x is productive 
Sx: x is a settlement
Tx: x has a trading port 
Wx: x is on the water
-Some settlements are not on the water.

Accepted Answer

The answer of use: t: Tortuga
Bx: x creates bricks 
Cx:...

provide a conterexample in a finite domain to each given invalid argument. -1. (∃x)(Gx • Hx) 2. (∃x)(Gx • Jx) 3. (∀x)(Jx ⊃ Kx) / (∃x)(Hx • Jx)

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates. -Abhishek loves ice cream and pizza.

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)(Ax • Bx) ⊃ (∀x) Dx 2. ∼Da / (∀x)(Ax ⊃ ∼Bx)

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)(Ax ⊃ Bx) ⊃ (∃x)Cx 2. (∃x)(Ax • ∼Bx) 3. (∀x)(Dx ⊃ Bx) / (∀x)(Dx ⊃ Cx)

use: t: Tortuga Bx: x creates bricks Cx: x is a city Nx: x is nicely placed Px: x is productive Sx: x is a settlement Tx: x has a trading port Wx: x is on the water -Either Tortuga is a city or it is not a settlement.

Translate each of the following sentences into predicate logic, using the given constants and predicates. -Farzona's dropping art history is a sufficient condition for her being unhappy. (f: Farzona; Ax: x drops art history; Ux: x is unhappy)

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates. -Some blankets are not soft.

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates. -Some cherries are red.

select the best translation into predicate logic. -Either Tortuga is a city or it is not a settlement.

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. -(Pf • Pg) • [(Rf • Rg) • (Sf • ∼Sg)]

1. ∼(∃x)Ax 2. (∀x)∼Ax ⊃ ∼(∃x)Bx -Which of the following propositions is derivable from the given premises in M?

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

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates. -If Carla works for an airline, then Darlene doesn't.

1. (∀x)[Ax ⊃ (Bx ⊃ Cx)] 2. ∼(∀x)(Bx ⊃ Dx) -Which of the following propositions is derivable in M from the given premises?

select the best English interpretation of the given statements of predicate logic. 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]

Translate each of the following sentences into predicate logic, using the given constants and predicates. -Some excellent doctors are not gentle. (Dx: x is a doctor; Ex: x is excellent; Gx: x is gentle)

Some websites have open comments which are not anonymous. Any website is either anonymous or requires a login. So something with open comments requires a login. -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

use: t: Tortuga Bx: x creates bricks Cx: x is a city Nx: x is nicely placed Px: x is productive Sx: x is a settlement Tx: x has a trading port Wx: x is on the water -Some settlements are not on the water.

Introducing Logic

Propositional Logic: Syntax and Semantic

Inference in Propositional Logic

Full First-Order Logic

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Exam 4: Monadic Predicate Logic