Question 1

refer to the following formula:
∼(∀x){(Ix • Jx) ⊃ [Kx ≡ (La • Lb)]}
-Which of the following variables in the formula are free?

Accepted Answer

A)  The x that follows the I. 
B)  The x that follows the J. 
C)  The x that follows the K. 
D)  All of the above. 
E)  None of the above. 
A)  The x that follows the I. 
B)  The x that follows the J. 
C)  The x that follows the K. 
D)  All of the above. 
E)  None of the above.

Question 2

1. (∀x)Ix $ \lor $ ∼(∃x)Hx
2. (∃x)Jx ⊃ ∼(∀x)Ix
3. Hc / (∀x)∼Jx
-Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

Accepted Answer

A)  Jx 
B)  (∀x)∼Jx 
C)  ∼(∀x)∼Jx 
D)  ∼(∀x)Jx 
E)  (∀x)Jx 
A)  Jx 
B)  (∀x)∼Jx 
C)  ∼(∀x)∼Jx 
D)  ∼(∀x)Jx 
E)  (∀x)Jx

Question 3

select the best translation into predicate logic.
-Only consistent rationalists are apriorists.

Accepted Answer

A)  (∀x)[(Cx • Rx)] ⊃ Ax] 
B)  (∀x)[Ax ≡ (Cx • Rx)] 
C)  (∀x)[(Rx $ \lor $ Ax) ⊃ Cx] 
D)  (∀x)[(Rx • Ax) ⊃ Cx] 
A)  (∀x)[(Cx • Rx)] ⊃ Ax] 
B)  (∀x)[Ax ≡ (Cx • Rx)] 
C)  (∀x)[(Rx $ \lor $ Ax) ⊃ Cx] 
D)  (∀x)[(Rx • Ax) ⊃ Cx]

Question 4

refer to the following formula: ∼(∀x){(Ix • Jx) ⊃ [Kx ≡ (La • Lb)]}
-Which variables are bound by the '(∀x)'?

Accepted Answer

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

Question 5

Translate each of the following sentences into predicate logic, using the given constants and predicates.
-Bonita doesn't study law; she's pre-med. (b: Bonita; Lx: x studies law; Mx: x is pre-med)

Accepted Answer

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

Question 6

provide a conterexample in a finite domain to each given invalid argument.
-1. (∀x)(Kx $ \lor $ Lx)
2. (∃x)∼Kx / (∀x)Lx

Accepted Answer

The answer of provide a conterexample in a finite domain...

Question 7

Translate each of the following sentences into predicate logic, using the given constants and predicates.
-If Tranh takes a sabbatical then neither she nor Minh will feel overworked. (m: Minh; t: Tranh; Ox: x feels overworked; Sx: x takes a sabbatical)

Accepted Answer

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

Question 8

provide a conterexample in a finite domain to each given invalid argument.
-1. (∃x)(Mx • Nx)
2. (∀x)(Ox ⊃ Mx) / (∀x)(Ox ⊃ Nx)

Accepted Answer

The answer of provide a conterexample in a finite domain...

Question 9

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 is the best translation into M of this argument?

Accepted Answer

A)  $\begin{array} { l } 
( \exists \mathrm { x } ) [ ( \mathrm { Mx } \cdot \mathrm { Ex } ) \cdot ( \mathrm { Lx } \vee \mathrm { Dx } ) ] \
\sim ( \forall \mathrm { x } ) ( \mathrm { Ex } \supset \mathrm { Dx } ) \quad \quad \quad \quad \quad / ( \exists \mathrm { x } ) ( \mathrm { Mx } \cdot \mathrm { Lx } ) \
\end{array}$ 
B)  $\begin{array} { l } 
( \exists \mathrm { x } ) [ ( \mathrm { Mx } \cdot \mathrm { Ex } ) \cdot ( \mathrm { Lx } \vee \mathrm { Dx } ) ] \
\sim ( \forall \mathrm { x } ) ( \mathrm { Ex } \supset \mathrm { Dx } ) \quad\quad \quad \quad \quad \quad  / ( \exists \mathrm { x } ) ( \mathrm { Mx } \supset \mathrm { Lx } ) \
\end{array}$ 
C)  $\begin{array} { l l } 
( \exists \mathrm { x } ) [ ( \mathrm { Mx } \vee \mathrm { Ex } ) \cdot ( \mathrm { Lx } \vee \mathrm { Dx } ) ] & \
( \forall \mathrm { x } ) ( \mathrm { Ex } \supset \sim \mathrm { Dx } ) & / ( \exists \mathrm { x } ) ( \mathrm { Mx } \cdot \mathrm { Lx } )
\end{array}$ 
D)  $\begin{array} { l } 
( \exists \mathrm { x } ) [ ( \mathrm { Mx } \cdot \mathrm { Ex } ) \vee ( \mathrm { Lx } \cdot \mathrm { Dx } ) ] \
( \forall \mathrm { x } ) ( \mathrm { Ex } \supset \sim \mathrm { Dx } ) \quad / ( \exists \mathrm { x } ) ( \mathrm { Mx } \supset \mathrm { Lx } ) \
\end{array}$ 
E)  $\begin{array} { l } 
( \exists \mathrm { x } ) [ ( \mathrm { Mx } \cdot \mathrm { Ex } ) \cdot ( \mathrm { Lx } \vee \mathrm { Dx } ) ] \
( \forall \mathrm { x } ) ( \mathrm { Ex } \supset \sim \mathrm { Dx } ) \quad / ( \exists \mathrm { x } ) ( \mathrm { Mx } \cdot \mathrm { Lx } ) \
\end{array}$ 
A)  $\begin{array} { l } 
( \exists \mathrm { x } ) [ ( \mathrm { Mx } \cdot \mathrm { Ex } ) \cdot ( \mathrm { Lx } \vee \mathrm { Dx } ) ] \
\sim ( \forall \mathrm { x } ) ( \mathrm { Ex } \supset \mathrm { Dx } ) \quad \quad \quad \quad \quad / ( \exists \mathrm { x } ) ( \mathrm { Mx } \cdot \mathrm { Lx } ) \
\end{array}$ 
B)  $\begin{array} { l } 
( \exists \mathrm { x } ) [ ( \mathrm { Mx } \cdot \mathrm { Ex } ) \cdot ( \mathrm { Lx } \vee \mathrm { Dx } ) ] \
\sim ( \forall \mathrm { x } ) ( \mathrm { Ex } \supset \mathrm { Dx } ) \quad\quad \quad \quad \quad \quad  / ( \exists \mathrm { x } ) ( \mathrm { Mx } \supset \mathrm { Lx } ) \
\end{array}$ 
C)  $\begin{array} { l l } 
( \exists \mathrm { x } ) [ ( \mathrm { Mx } \vee \mathrm { Ex } ) \cdot ( \mathrm { Lx } \vee \mathrm { Dx } ) ] & \
( \forall \mathrm { x } ) ( \mathrm { Ex } \supset \sim \mathrm { Dx } ) & / ( \exists \mathrm { x } ) ( \mathrm { Mx } \cdot \mathrm { Lx } )
\end{array}$ 
D)  $\begin{array} { l } 
( \exists \mathrm { x } ) [ ( \mathrm { Mx } \cdot \mathrm { Ex } ) \vee ( \mathrm { Lx } \cdot \mathrm { Dx } ) ] \
( \forall \mathrm { x } ) ( \mathrm { Ex } \supset \sim \mathrm { Dx } ) \quad / ( \exists \mathrm { x } ) ( \mathrm { Mx } \supset \mathrm { Lx } ) \
\end{array}$ 
E)  $\begin{array} { l } 
( \exists \mathrm { x } ) [ ( \mathrm { Mx } \cdot \mathrm { Ex } ) \cdot ( \mathrm { Lx } \vee \mathrm { Dx } ) ] \
( \forall \mathrm { x } ) ( \mathrm { Ex } \supset \sim \mathrm { Dx } ) \quad / ( \exists \mathrm { x } ) ( \mathrm { Mx } \cdot \mathrm { Lx } ) \
\end{array}$

Question 10

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 three sentences which uses all three predicates and at least three different constants.

Accepted Answer

The theory can be constructed as follows

Question 11

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)[Fx • (Gx • Hx)]
2. ∼(∃x)(Ix • ∼Fx) / (∀x)[Ix ⊃ (∼Gx $ \lor $ ∼Hx)]

Accepted Answer

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

Question 12

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
-1. (∃x)(Jx • ∼Kx) ⊃ (∃x)(Lx • Mx)
2) (∃x)(Jx • Lx)
3) ∼(∃x)(Jx • Kx) / ∼(∀x)(Lx ⊃ Mx)

Accepted Answer

A)  Valid 
B)  Invalid. Counterexample in a domain of 1 member, in which:
Ja: True Ka: False La: True Ma: True 
C)  Invalid. Counterexample in a domain of 1 member, in which:
Ja: False Ka: False La: True Ma: True 
D)  Invalid. Counterexample in a domain of 1 member, in which:
Ja: True Ka: True La: True Ma: True 
E)  Invalid. Counterexample in a domain of 1 member, in which:
Ja: True Ka: False La: False Ma: False 
A)  Valid 
B)  Invalid. Counterexample in a domain of 1 member, in which:
Ja: True Ka: False La: True Ma: True 
C)  Invalid. Counterexample in a domain of 1 member, in which:
Ja: False Ka: False La: True Ma: True 
D)  Invalid. Counterexample in a domain of 1 member, in which:
Ja: True Ka: True La: True Ma: True 
E)  Invalid. Counterexample in a domain of 1 member, in which:
Ja: True Ka: False La: False Ma: False

Question 13

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

Accepted Answer

A)  Hx • Gx 
B)  Hn • Gn 
C)  Fn ⊃ ∼Gx 
D)  Fx ⊃ ∼Gn 
E)  Hn 
A)  Hx • Gx 
B)  Hn • Gn 
C)  Fn ⊃ ∼Gx 
D)  Fx ⊃ ∼Gn 
E)  Hn

Question 14

Translate each of the following sentences into predicate logic, using the given constants and predicates.
-Some blankets are not soft. (Bx: x is a blanket; Sx: x is soft)

Accepted Answer

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

Question 15

select the best translation into predicate logic.
-Berkeley is an empiricist and Hume is not an apriorist.

Accepted Answer

A)  Eb • Ah 
B)  ∼Eb • ∼Ah 
C)  Eb • ∼Ah 
D)  (∃x)(Ex • ∼Ax) 
A)  Eb • Ah 
B)  ∼Eb • ∼Ah 
C)  Eb • ∼Ah 
D)  (∃x)(Ex • ∼Ax)

Question 16

Translate each of the following sentences into predicate logic, using the given constants and predicates.
-No red flowers are in the garden. (Fx: x is a flower; Gx: x is in the garden; Rx: x is red)

Accepted Answer

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

Question 17

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}
-(∃x)(Ax • Bx) 
(∃x)(Ax • ∼Bx) 
(∀x)[Ax ⊃ (Bx $ \lor $ Cx)]
(∃x)Ax ⊃ (∃x)(Ax • Bx)

Accepted Answer

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

Question 18

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)  Eb • ∼Ea Oa • ∼Eb Pb • ∼Pc 
B)  Eb • ∼Ea Oa • Eb Pb • Pc 
C)  Eb • ∼Ea
∼Oa • Eb Pb • ∼Pc 
D)  ∼Eb • ∼Ea
Oa • Eb Pb • ∼Pc 
E)  Eb • ∼Ea Oa • Eb Pb • ∼Pc 
A)  Eb • ∼Ea Oa • ∼Eb Pb • ∼Pc 
B)  Eb • ∼Ea Oa • Eb Pb • Pc 
C)  Eb • ∼Ea
∼Oa • Eb Pb • ∼Pc 
D)  ∼Eb • ∼Ea
Oa • Eb Pb • ∼Pc 
E)  Eb • ∼Ea Oa • Eb Pb • ∼Pc

Question 19

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

Accepted Answer

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

Question 20

select a counterexample for the given invalid argument.
-1. (∃x)(Hx • Ix)
2) (∃x)(Hx • ∼Ix)
3) (∀x)(Jx ⊃ Ix) / (∀x)(Jx ⊃ Hx)

Accepted Answer

A)  Counterexample in a domain of 3 members, in which: Ha: True Ia: True Ja: True
Hb: True Ib: False Jb: False
Hc: False Ic: True Jc: True 
B)  Counterexample in a domain of 3 members, in which: Ha: False Ia: True Ja: True
Hb: True Ib: True Jb: False
Hc: False Ic: True Jc: False 
C)  Counterexample in a domain of 3 members, in which: Ha: True Ia: True Ja: True
Hb: True Ib: False Jb: False
Hc: True Ic: True Jc: True 
D)  Counterexample in a domain of 3 members, in which: Ha: False Ia: True Ja: False
Hb: True Ib: False Jb: True
Hc: False Ic: True Jc: True 
E)  Counterexample in a domain of 3 members, in which: Ha: True Ia: False Ja: True
Hb: False Ib: False Jb: False
Hc: False Ic: False Jc: False 
A)  Counterexample in a domain of 3 members, in which: Ha: True Ia: True Ja: True
Hb: True Ib: False Jb: False
Hc: False Ic: True Jc: True 
B)  Counterexample in a domain of 3 members, in which: Ha: False Ia: True Ja: True
Hb: True Ib: True Jb: False
Hc: False Ic: True Jc: False 
C)  Counterexample in a domain of 3 members, in which: Ha: True Ia: True Ja: True
Hb: True Ib: False Jb: False
Hc: True Ic: True Jc: True 
D)  Counterexample in a domain of 3 members, in which: Ha: False Ia: True Ja: False
Hb: True Ib: False Jb: True
Hc: False Ic: True Jc: True 
E)  Counterexample in a domain of 3 members, in which: Ha: True Ia: False Ja: True
Hb: False Ib: False Jb: False
Hc: False Ic: False Jc: False

refer to the following formula: ∼(∀x){(Ix • Jx) ⊃ [Kx ≡ (La • Lb)]} -Which of the following variables in the formula are free?

1. (∀x)Ix $\lor$ ∼(∃x)Hx 2. (∃x)Jx ⊃ ∼(∀x)Ix 3. Hc / (∀x)∼Jx -Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

select the best translation into predicate logic. -Only consistent rationalists are apriorists.

refer to the following formula: ∼(∀x){(Ix • Jx) ⊃ [Kx ≡ (La • Lb)]} -Which variables are bound by the '(∀x)'?

Translate each of the following sentences into predicate logic, using the given constants and predicates. -Bonita doesn't study law; she's pre-med. (b: Bonita; Lx: x studies law; Mx: x is pre-med)

provide a conterexample in a finite domain to each given invalid argument. -1. (∀x)(Kx $\lor$ Lx) 2. (∃x)∼Kx / (∀x)Lx

Translate each of the following sentences into predicate logic, using the given constants and predicates. -If Tranh takes a sabbatical then neither she nor Minh will feel overworked. (m: Minh; t: Tranh; Ox: x feels overworked; Sx: x takes a sabbatical)

provide a conterexample in a finite domain to each given invalid argument. -1. (∃x)(Mx • Nx) 2. (∀x)(Ox ⊃ Mx) / (∀x)(Ox ⊃ Nx)

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 is the best translation into M of this argument?

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)[Fx • (Gx • Hx)] 2. ∼(∃x)(Ix • ∼Fx) / (∀x)[Ix ⊃ (∼Gx $\lor$ ∼Hx)]

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample. -1. (∃x)(Jx • ∼Kx) ⊃ (∃x)(Lx • Mx) 2) (∃x)(Jx • Lx) 3) ∼(∃x)(Jx • Kx) / ∼(∀x)(Lx ⊃ Mx)

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

Translate each of the following sentences into predicate logic, using the given constants and predicates. -Some blankets are not soft. (Bx: x is a blanket; Sx: x is soft)

select the best translation into predicate logic. -Berkeley is an empiricist and Hume is not an apriorist.

Translate each of the following sentences into predicate logic, using the given constants and predicates. -No red flowers are in the garden. (Fx: x is a flower; Gx: x is in the garden; Rx: x is red)

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

select a counterexample for the given invalid argument. -1. (∃x)(Hx • Ix) 2) (∃x)(Hx • ∼Ix) 3) (∀x)(Jx ⊃ Ix) / (∀x)(Jx ⊃ Hx)

Introducing Logic

Propositional Logic: Syntax and Semantic

Inference in Propositional Logic

Full First-Order Logic

Filters

Exam 4: Monadic Predicate Logic

refer to the following formula: ∼(∀x){(Ix • Jx) ⊃ [Kx ≡ (La • Lb)]} -Which of the following variables in the formula are free?

1. (∀x)Ix ∨ \lor ∨ ∼(∃x)Hx 2. (∃x)Jx ⊃ ∼(∀x)Ix 3. Hc / (∀x)∼Jx -Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

select the best translation into predicate logic. -Only consistent rationalists are apriorists.

refer to the following formula: ∼(∀x){(Ix • Jx) ⊃ [Kx ≡ (La • Lb)]} -Which variables are bound by the '(∀x)'?

Translate each of the following sentences into predicate logic, using the given constants and predicates. -Bonita doesn't study law; she's pre-med. (b: Bonita; Lx: x studies law; Mx: x is pre-med)

provide a conterexample in a finite domain to each given invalid argument. -1. (∀x)(Kx ∨ \lor ∨ Lx) 2. (∃x)∼Kx / (∀x)Lx

Translate each of the following sentences into predicate logic, using the given constants and predicates. -If Tranh takes a sabbatical then neither she nor Minh will feel overworked. (m: Minh; t: Tranh; Ox: x feels overworked; Sx: x takes a sabbatical)

provide a conterexample in a finite domain to each given invalid argument. -1. (∃x)(Mx • Nx) 2. (∀x)(Ox ⊃ Mx) / (∀x)(Ox ⊃ Nx)

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 is the best translation into M of this argument?

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)[Fx • (Gx • Hx)] 2. ∼(∃x)(Ix • ∼Fx) / (∀x)[Ix ⊃ (∼Gx ∨ \lor ∨ ∼Hx)]

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample. -1. (∃x)(Jx • ∼Kx) ⊃ (∃x)(Lx • Mx) 2) (∃x)(Jx • Lx) 3) ∼(∃x)(Jx • Kx) / ∼(∀x)(Lx ⊃ Mx)

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

Translate each of the following sentences into predicate logic, using the given constants and predicates. -Some blankets are not soft. (Bx: x is a blanket; Sx: x is soft)

select the best translation into predicate logic. -Berkeley is an empiricist and Hume is not an apriorist.

Translate each of the following sentences into predicate logic, using the given constants and predicates. -No red flowers are in the garden. (Fx: x is a flower; Gx: x is in the garden; Rx: x is red)

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

select a counterexample for the given invalid argument. -1. (∃x)(Hx • Ix) 2) (∃x)(Hx • ∼Ix) 3) (∀x)(Jx ⊃ Ix) / (∀x)(Jx ⊃ Hx)

Introducing Logic

Propositional Logic: Syntax and Semantic

Inference in Propositional Logic

Full First-Order Logic

Filters

1. (∀x)Ix $\lor$ ∼(∃x)Hx 2. (∃x)Jx ⊃ ∼(∀x)Ix 3. Hc / (∀x)∼Jx -Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

provide a conterexample in a finite domain to each given invalid argument. -1. (∀x)(Kx $\lor$ Lx) 2. (∃x)∼Kx / (∀x)Lx

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)[Fx • (Gx • Hx)] 2. ∼(∃x)(Ix • ∼Fx) / (∀x)[Ix ⊃ (∼Gx $\lor$ ∼Hx)]

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