Question 1

1. (∃x)[Hx • (Ix $ \lor $ Jx)]
2. (∀x)(Kx ⊃ ∼Ix)
3. (∀x)(Hx ⊃ Kx)
-Which of the following propositions is an immediate (one-step) consequence in M of the given premises?
A) Hc • (Ic $ \lor $ Jc)
B) Ks ⊃ ∼Ix
C) Hx ⊃ Kw
D) Hx • (Ix $ \lor $ Jx)
E) Ha • (Ib $ \lor $ Jc)

Accepted Answer

The first premise directly states that there exists an x such that Hx and (Ix or Jx) are true, which is exactly what choice A expresses for a specific instance 'c'.

Question 2

1. (∃x)[Hx • (Ix $ \lor $ Jx)]
2. (∀x)(Kx ⊃ ∼Ix)
3. (∀x)(Hx ⊃ Kx)
-Which of the following propositions is derivable from the given premises in M?
A) (∀x)Jx
B) (∃x)Jx
C) (∀x)Ix
D) (∃x)Ix
E) (∀x)(Ix • ∼Jx)

Accepted Answer

From premise 1, we know that there exists an x such that Hx is true and either Ix or Jx is true. From premise 3, we know that if Hx is true, then Kx is true. Thus, we have (∃x)Kx ∨ Jx. From premise 2, we know that for all x, if Kx is true, then ∼Ix is true. Therefore, we can conclude that (∃x)Jx must be true, since if there were no x for which Jx is true, then all Kx would be true, and therefore all Ix would be false, contradicting the existence of an x for which Ix is true in premise 1.

Question 3

1. (∃x)(Mx • Ox) ⊃ (∃x)Nx
2. (∃x)(Px • Mx)
3. (∀x)(∼Px $ \lor $ Ox)
-Which of the following propositions is an immediate (one-step) consequence in M of the given premises?
A) Px • Mx
B) Px $ \lor $ Ox
C) (Ma • Ob) ⊃ (∃x)Nx
D) Pc • Mc
E) ∼Pn $ \lor $ Ox

Accepted Answer

The answer of 1. (∃x)(Mx • Ox) ⊃ (∃x)Nx
2. (∃x)(Px...

Question 4

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

Accepted Answer

1. (∃x)(Mx • Ox) ⊃ (∃x)Nx  (Premise)
2. (∃x)(Px • Mx)  (Premise)
3. (∀x)(∼Px V Ox)  (Premise)

From premise 2, we know that there exists at least one object x that is both P and M.
Let's assume that this object is a. So we have:

4. Pa • Ma (from 2 with existential instantiation)

From premise 1, we have:

5. (Ma • Oa) ⊃ (∃x)Nx (from 1)

Substituting a for x in the antecedent of 5, we get:

6. (Ma • Oa) ⊃ (∃x)Nx (instantiated with a)

From premise 3, we have:

7. (∀x)(∼Px V Ox)  (Premise)

Substituting a for x, we get:

8. ∼Pa V Oa (instantiated with a)

Using simplification, we can obtain:

9. Oa (from 4 and 8)

Using modus ponens with 6 and 9, we get:

10. (∃x)Nx (from 5 and 9)

Therefore, the derivable proposition is (∃x)Nx.

Question 5

1. (∀x)Ix ⊃ (∀x)Kx
2. (∀x)[Jx • (Ix $ \lor $ Lx)]
3. (∀x)(Jx ⊃ ∼Lx)
-Which of the following propositions is an immediate (one-step) consequence in M of the given premises?
A) Jx • (Iv $ \lor $ Lv)
B) Ix
C) Jx • (Ix $ \lor $ Lx)]
D) Ix ⊃ Kx
E) Jx ⊃ ∼Lp

Accepted Answer

The answer of 1. (∀x)Ix ⊃ (∀x)Kx
2. (∀x)[Jx • (Ix...

Question 6

1. (∀x)Ix ⊃ (∀x)Kx
2. (∀x)[Jx • (Ix $ \lor $ Lx)]
3. (∀x)(Jx ⊃ ∼Lx)
-Which of the following propositions is derivable from the given premises in M?
A) (∃x)∼Jx
B) (∃x)Lx
C) (∀x)Lx
D) (∀x)Kx
E) (∀x)∼Jx

Accepted Answer

From the premises, we can derive that (∀x)Kx. Here's a brief explanation: 1. From the second premise, we know that for all x, Jx and (Ix or Lx). 2. The third premise tells us that for all x, if Jx then not Lx, which means for all x, we have Jx and Ix (since Lx leads to a contradiction with Jx).3. Given that for all x, Ix (from step 2), and the first premise states that if for all x, Ix then for all x, Kx, we can conclude that (∀x)Kx.

Question 7

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 is the best translation into M of this argument?
A) (∃x)[Wx • (Ox • ∼Ax)]
(∀x)[Wx ⊃ (Ax $ \lor $ Lx)] / (∀x)(Ox • Lx)
B) (∃x)[Wx • (Ox • ∼Ax)]
(∀x)[Wx ⊃ (Ax $ \lor $ Lx)] / (∃x)(Ox • Lx)
C) (∃x)[Wx • ∼(Ox • Ax)]
(∀x)(Ax $ \lor $ Lx) / (∃x)(Ox • Lx)
D) (∃x)[Wx • (Ox • ∼Ax)]
(∀x)[Wx ⊃ (Ax • Lx)] / (∃x)(Ox • Lx)
E) (∃x)[Wx • (Ox • ∼Ax)]
(∀x)(Ax $ \lor $ Lx) / (∃x)(Ox • Lx)

Accepted Answer

This argument can be translated as:
There exists a website with open comments which are not anonymous (Wx • (Ox • ∼Ax)). For any website, it either requires a login or is anonymous (Wx ⊃ (Ax $ \lor $ Lx)). Therefore, there exists a website that requires a login and has open comments ((∃x)(Ox • Lx)). Choice B matches this translation.

Question 8

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

Accepted Answer

- The given premises establish that websites with open comments require login, which can be expressed as Wx ⊃ Lx.
- As per the options provided, only option C includes the necessary implication (Wx ⊃ ...) and does not include any extraneous information. Thus, C is the immediate consequence in M of the given premises.

Question 9

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 claims can also be derived from the premises of this argument?
A) Ab $ \lor $ Lb
B) (∀x)(Ax $ \lor $ Lx)
C) (∀x)(Ox • Lx)
D) (∃x)Ax
E) (∃x)∼Wx

Accepted Answer

The answer of Some websites have open comments which are...

Question 10

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?
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) $\begin{array} { l } 
( \exists \mathrm { x } ) [ ( \mathrm { Mx } \cdot \mathrm { Ex } ) \cdot ( \mathrm { Lx } \vee \mathrm { Dx } ) ] \
( \forall \mathrm { x } ) ( \mathrm { Ex } \supset \sim \mathrm { Dx } ) \quad / ( \exists \mathrm { x } ) ( \mathrm { Mx } \cdot \mathrm { Lx } ) \
\end{array}$

Accepted Answer

Option E accurately translates the argument. The premises state that some materialist empiricists are either libertarians or hard determinists, and no empiricist is a hard determinist, leading to the conclusion that some materialists are libertarians. This is correctly represented in E with existential quantifiers for the existence claims and a universal quantifier negating the possibility of empiricists being hard determinists.

Question 11

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 propositions is an immediate (one-step) consequence in M of the given premises?
A) (Mx • Ex) • (Lx $ \lor $ Dx)
B) (Mc • Ec) • (Lc $ \lor $ Dc)
C) Ex ⊃ ∼Dc
D) En ⊃ ∼Dx
E) Mc • Ec

Accepted Answer

The answer of Some materialist empiricists are either libertarians or...

Question 12

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

Accepted Answer

The argument states that some materialists are libertarians and that no empiricist is a hard determinist. This allows us to infer that there exists at least one person who is both a materialist (M) and a libertarian (L), and is not a hard determinist (∼D). This can be represented as (∃x)[(Mx • Ex) • (Lx • ∼Dx)], which is option A.

Question 13

1. (∀x)(Jx ⊃ Kx)
2. ∼(∀x)Kx
-Which of the following propositions is an immediate (one-step) consequence in M of the given premises?
A) (∀x)∼Kx
B) Ja ⊃ Kb
C) ∼(∃x)Kx
D) (∃x)∼Kx
E) ∼(∃x)∼Kx

Accepted Answer

From premise 2 and negation introduction, we can infer (∃x)∼Kx, which is choice D. We cannot infer A or C because they require universal quantification over Kx, which is not given in the premises. Option B involves a and b, which are not present in the premises. Option E introduces double negation, which is not a valid inference from the premises.

Question 14

1. (∀x)(Jx ⊃ Kx)
2. ∼(∀x)Kx
-Which of the following propositions is derivable from the given premises in M?
A) (∃x)Kx
B) ∼(∀x)∼Jx
C) (∀x)∼Jx
D) ∼(∀x)Jx
E) ∼(∃x)Kx

Accepted Answer

From premise 1, we can infer that if something is J, then it is also K. So, if everything is K, then everything is also J. But premise 2 tells us that not everything is K, which means there must exist something that is not K. From this, we can infer that there must exist something that is not J, represented by choice (D), ∼(∀x)Jx.

Question 15

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

Accepted Answer

The first premise states that there does not exist an x such that Ax is true. The second premise states that if for all x, Ax is not true, then there does not exist an x such that Bx is true. From these, an immediate consequence is that if there does not exist an x such that Ax is true (which is given in the first premise), then there does not exist an x such that Bx is true, which matches option E. This follows directly from the structure of the second premise when considering the truth of the first premise.

Question 16

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

Accepted Answer

The answer of 1. ∼(∃x)Ax
2. (∀x)∼Ax ⊃ ∼(∃x)Bx
-Which of the...

Question 17

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?
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

Accepted Answer

The answer of 1. (∃x)(Ax • Bx) ⊃ (∀x)Dx
2. ∼Da
-Which...

Question 18

1. (∃x)(Ax • Bx) ⊃ (∀x)Dx
2. ∼Da
-Which of the following propositions is derivable from the given premises in M?
A) (∀x)(Ax • Dx)
B) (∃x)(Ax • Dx)
C) (∀x)(Ax ⊃ Bx)
D) (∀x)(Ax ⊃ ∼Bx)
E) ∼(∃x)Dx

Accepted Answer

The answer of 1. (∃x)(Ax • Bx) ⊃ (∀x)Dx
2. ∼Da
-Which...

Question 19

1. (∃x)(Hx • ∼Ix)
2. (∀x)(Hx ⊃ Jx)
3. ∼(∃x)(Jx • Kx)
-Which of the following propositions is an immediate (one-step) consequence in M of the given premises?
A) (∀x)∼(Jx • Kx)
B) ∼(Jx • Kx)
C) Jx • Kx
D) Hx • ∼Ix
E) ∼(∀x)∼(Jx • Kx)

Accepted Answer

BA) From premise 3, we can infer that for all x, it is not the case that both Jx and Kx are true, which is a restatement of the negation of the existential quantifier in premise 3.B) Premise 3 directly states that there does not exist an x such that both Jx and Kx are true, which is the same as saying it is not the case that Jx and Kx for any particular instance of x.

Question 20

1. (∃x)(Hx • ∼Ix)
2. (∀x)(Hx ⊃ Jx)
3. ∼(∃x)(Jx • Kx)
-Which of the following propositions is derivable from the given premises in M?
A) (∃x)Ix
B) (∀x)∼(Ix $ \lor $ Kx)
C) (∃x) ∼Hx
D) (∀x)∼(Ix • Kx)
E) (∃x)∼(Ix $ \lor $ Kx)

Accepted Answer

From premise 1, we can derive:
- (∃x)Hx (by existential instantiation)
From premise 2, we can derive:
- (∀x)Jx ∨ ∼Hx (by contrapositive)
From premise 3, we can derive:
- ∼(∃x)Jx ∨ ∼(∃x)Kx (by De Morgan's and quantifier negation)
- (∀x)∼Jx ∨ (∀x)∼Kx (by quantifier negation)

To derive (∃x)∼(Ix $ \lor $ Kx), we can use proof by contradiction. Assume ¬(∃x)∼(Ix $ \lor $ Kx), i.e., (∀x)(Ix $ \lor $ Kx). Then, we can derive:
- (∀x)∼∼(Ix ⊃ Kx) (by contrapositive)
- (∀x)(Ix ⊃ Kx) (by double negation)
- (∀x)(∼Kx ⊃ ∼Ix) (by contrapositive)
- (∀x)(Jx ⊃ ∼Ix) (by premise 2 and transitivity)
- (∀x)∼Jx (by proof by contradiction and disjunctive syllogism with premise 3)
- (∀x)∼(Hx ⊃ Jx) (by contrapositive)
- (∀x)(∼Jx ⊃ ∼Hx) (by contrapositive)
- (∀x)∃y∼Hy (by premise 1 and universal instantiation)
- (∀x)(∃y∼Hy ⊃ ∃y∼Ix) (by universal instantiation)
- (∀x)∃y∼Ix (by reasoning by cases, since either ∼Hy or ∃y∼Ix must hold for each x)
- ∃x∃y∼Ix (by existential instantiation)
- (∃x)∼(Ix ⊃ Kx) ∧ (∃x)∃y∼Ix (by conjunction)
- (∃x)∼((Ix ⊃ Kx) ⊃ (∀y)∼Iy) (by existential generalization)
- (∃x)∼((∀y)(Ix ⊃ Kx) ⊃ (∀y)∼Iy) (by universal instantiation)
- (∃x)∼(∀y)((Ix ⊃ Kx) ⊃ ∼Iy) (by quantifier exchange)
- (∃x)(∃y)((Ix ⊃ Kx) ∧ Iy) (by De Morgan's and quantifier negation)
- (∃x)∃y(Ix ∧ Iy ∧ Kx) (by reasoning by cases, since either Iy or Kx must hold for each x and y)
- (∃x)Ix ∧ (∃y)(Iy ∧ Kx) (by existential generalization)
- (∃x)Ix (by existential instantiation)

This leads to a contradiction with our assumption. Therefore, we can conclude (∃x)∼(Ix $ \lor $ Kx).

Quiz 4: Monadic Predicate Logic

1. (∃x) [Hx • (Ix $\lor$ Jx) ] 2. (∀x) (Kx ⊃ ∼Ix) 3. (∀x) (Hx ⊃ Kx) -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

1. (∃x) [Hx • (Ix $\lor$ Jx) ] 2. (∀x) (Kx ⊃ ∼Ix) 3. (∀x) (Hx ⊃ Kx) -Which of the following propositions is derivable from the given premises in M?

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

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?

1. (∀x) Ix ⊃ (∀x) Kx 2. (∀x) [Jx • (Ix $\lor$ Lx) ] 3. (∀x) (Jx ⊃ ∼Lx) -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

1. (∀x) Ix ⊃ (∀x) Kx 2. (∀x) [Jx • (Ix $\lor$ Lx) ] 3. (∀x) (Jx ⊃ ∼Lx) -Which of the following propositions is derivable from the given premises in M?

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 is the best translation into M of this argument?

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?

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 claims can also be derived from the premises of this argument?

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?

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 propositions is an immediate (one-step) consequence in M of the given premises?

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?

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

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

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?

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?

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

1. (∃x) (Hx • ∼Ix) 2. (∀x) (Hx ⊃ Jx) 3. ∼(∃x) (Jx • Kx) -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

1. (∃x) (Hx • ∼Ix) 2. (∀x) (Hx ⊃ Jx) 3. ∼(∃x) (Jx • Kx) -Which of the following propositions is derivable from the given premises in M?