Question 1

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

Accepted Answer

The answer of 1. (∃x)∼Ix ⊃ (∀x)(Jx $ \lor $...

Question 2

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

Accepted Answer

The premises suggest that there exists an x such that Ix is false and Jb is false. From the first premise, if there exists an x such that Ix is false, then for all x, Jx or Kx must be true. Since Jb is false, Kb must be true.

Question 3

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

Accepted Answer

The proposition ∼(∀x)Ix $ \lor $ ∼(∃x)∼Jx is an immediate consequence because it is logically equivalent to the given premise 3, ∼(∀x)Ix $ \lor $ (∀x)Jx, by the negation of the existential quantifier.

Question 4

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

Accepted Answer

Using universal instantiation on premise 1, we have:
(Ja ⊃ Ka) 
Using modus tollens on premise 2, we have:
∃x ∼Kx (from negating universal quantifier) 
∼Ka
Using disjunctive syllogism on premise 3, we have:
∼Ia 
Therefore, (∃x)∼Ix is derivable.

Question 5

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?
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)

Accepted Answer

Option B is correct because it applies the logical equivalence between an existential quantifier and the negation of a universal quantifier. Given the premise (∃x)(Gx • ∼Hx) ≡ (∃x)(Hx • ∼Ix), negating both sides of the equivalence and applying De Morgan's laws gives us the equivalent statement in option B.

Question 6

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

Accepted Answer

From the first premise, we know there exists an x such that Gx and not Hx, and there exists an x such that Hx and not Ix. The second premise states that for all x, if Hx then Ix. To derive a conclusion, we notice that the existence claims do not directly lead to a universal claim about all x without additional premises. However, by understanding that if Hx then Ix (from the second premise), and considering the structure of the first premise, we can infer that if something is Gx, it might lead to a contradiction unless it also leads to Hx under some conditions to avoid the contradiction implied by the existence statements. Thus, the most directly derivable statement given the premises and avoiding contradictions is that for all x, if Gx then Hx, which is choice D. This is because the second premise ensures that if Hx is true, then Ix must also be true, indirectly suggesting a relationship between Gx and Hx to maintain logical consistency without directly asserting the existence of an x that is both Gx and Ix, which is not supported by the given premises.

Question 7

1. ∼(∃x)[Fx • (Gx • Hx)]
2. ∼(∃x)(Ix • ∼Fx)
-Which of the following propositions is an immediate (one-step) consequence in M of the given premises?
A) (∀x)∼[Fx • (Gx • Hx)]
B) Fx • (Gx • Hx)
C) Fa • (Ga • Ha)
D) (∃x)∼[Fx • (Gx • Hx)]
E) ∼(∀x)[Fx • (Gx • Hx)]

Accepted Answer

The first premise is a negation of an existential quantifier, which directly implies the universal negation of the proposition within the quantifier, making choice A correct.

Question 8

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

Accepted Answer

To derive proposition C, we can use universal instantiation to obtain (∼Gy • ∼Hy) for any object y, and then use modus ponens with proposition 2 to obtain Iy, and finally use conditional introduction to obtain the desired result for any object y. Therefore, (∀x)[Ix ⊃ (∼Gx $ \lor $ ∼Hx)] is derivable from the given premises. None of the other answer choices can be derived from the premises.

Question 9

If anything is stuck in traffic, Bruno will be mad if it is late. Something is neither annoyed nor not late. If Bruno is mad, then something is annoyed. So something is not stuck in traffic.
-Which of the following is the best translation into M of this argument?
A) $\begin{array} { l } 
( \forall \mathrm { x } ) [ \mathrm { Sx } \supset ( \mathrm { Mb } \supset \mathrm { Lx } ) ] \
( \forall \mathrm { x } ) \sim ( \mathrm { Ax } \vee \sim \mathrm { Lx } ) \
\mathrm { Mb } \supset ( \exists \mathrm { x } ) \mathrm { Ax } \quad&/(\exists x) \sim S x
\end{array}$
B) $\begin{array} { l } 
( \exists \mathrm { x } ) \mathrm { Sx } \supset ( \mathrm { Mb } \supset \mathrm { Lx } ) \
( \forall \mathrm { x } ) \sim ( \mathrm { Ax } \vee \sim \mathrm { Lx } ) \
\mathrm { Mb } \supset ( \exists \mathrm { x } ) \mathrm { Ax } \quad / ( \exists \mathrm { x } ) \sim \mathrm { Sx }
\end{array}$
C) $\begin{array} { l } 
( \exists \mathrm { x } ) \mathrm { Sx } \supset ( \mathrm { Lx } \supset \mathrm { Mb } ) \
( \forall \mathrm { x } ) \sim ( \mathrm { Ax } \vee \sim \mathrm { Lx } ) \
\mathrm { Mb } \supset ( \exists \mathrm { x } ) \mathrm { Ax } \quad \quad ( \exists \mathrm { x } ) \sim \mathrm { Sx }
\end{array}$
D) $\begin{array} { l } 
( \forall \mathrm { x } ) [ \mathrm { Sx } \supset ( \mathrm { Lx } \supset \mathrm { Mb } ) ] \
( \forall \mathrm { x } ) \sim ( \mathrm { Ax } \vee \sim \mathrm { Lx } ) \
\mathrm { Mb } \supset ( \exists \mathrm { x } ) \mathrm { Ax } \quad \quad \quad \quad / ( \exists \mathrm { x } ) \sim \mathrm { Sx }
\end{array}$
E) $\begin{array} { l } 
( \forall \mathrm { x } ) \mathrm { Sx } \supset ( \mathrm { Lx } \supset \mathrm { Mb } ) \
( \forall \mathrm { x } ) \sim ( \mathrm { Ax } \vee \sim \mathrm { Lx } ) \
\mathrm { Mb } \supset ( \exists \mathrm { x } ) \mathrm { Ax } \quad / ( \exists \mathrm { x } ) \sim \mathrm { Sx }
\end{array}$

Accepted Answer

The correct translation involves universal quantifiers for the general statements about any 'x' and captures the logical structure of the argument accurately, including the implication relationships and the negation of being stuck in traffic as the conclusion.

Question 10

If anything is stuck in traffic, Bruno will be mad if it is late. Something is neither annoyed nor not late. If Bruno is mad, then something is annoyed. So something is not stuck in traffic.
-Which of the following propositions is an immediate (one-step) consequence in M of the given premises?
A) Mb ⊃ Ab
B) Mb ⊃ Ae
C) (∀x) ∼(Ax • ∼∼Lx)
D) (∃x)(Ax $ \lor $ ∼Lx)
E) ∼ (∃x)(∼Ax • ∼∼Lx)

Accepted Answer

The first premise can be translated as "If Bruno is mad, then the thing is annoyed." Using contraposition, we can rewrite this as "If the thing is not annoyed, then Bruno is not mad." 
The second premise states that "Something is neither annoyed nor not late", which can be rewritten as "Something is not annoyed and not not-late", or simply "Something is not annoyed and is not late."
Combining these two statements gives us "Something is not annoyed and Bruno is not mad", which can be expressed using logic symbols as ∼(A • ∼Mb). Using De Morgan's laws, this can be rewritten as (∀x) ∼(Ax • ∼∼Lx). Therefore, C is an immediate consequence of the given premises.

Question 11

If anything is stuck in traffic, Bruno will be mad if it is late. Something is neither annoyed nor not late. If Bruno is mad, then something is annoyed. So something is not stuck in traffic.
-Which of the following claims can also be derived from the premises of this argument?
A) (∃x)Lx
B) ∼Sb $ \lor $ ∼Lb
C) ∼Mb
D) All of the above.
E) None of the above.

Accepted Answer

Given the premises, we can deduce that if Bruno is mad, it implies something is annoyed. However, it's stated that something is neither annoyed nor not late, which contradicts the possibility of Bruno being mad based on the initial condition. Since Bruno being mad leads to something being annoyed, and we know something is not annoyed, Bruno cannot be mad. This also implies that the something is not stuck in traffic, as its lateness would make Bruno mad. Therefore, all of the options provided can be derived from the premises: A) Something is late, B) Something is not stuck in traffic or not late, and C) Bruno is not mad.

Question 12

If some journalists are not respectable, then all journalists on the web have work to do. Some journalists on the web lack assignments. It is not the case that something without an assignment has work to do. So all journalists are respectable.
-Which of the following is the best translation into M of this argument?
A) (∃x)(Jx • ∼Rx) ⊃ (∀x)[(Jx • Ix) ⊃ Wx] (∃x)[(Jx • Ix) • ∼Ax]
(∃x)(∼Ax • Wx) / (∀x)(Jx ⊃ Rx)
B) (∃x)(Jx • ∼Rx) ⊃ (∀x)[(Jx • Ix) ⊃ Wx] (∃x)[(Jx • Ix) • ∼Ax]
∼(∃x)(∼Ax • Wx) / (∀x)(Jx ⊃ Rx)
C) ∼(∃x)(Jx • Rx) ⊃ (∀x)[(Jx • Ix) ⊃ Wx] (∃x)[(Jx • Ix) • ∼Ax]
∼(∃x)(∼Ax • Wx) / (∀x)(Jx ⊃ Rx)
D) ∼(∃x)(Jx • ∼Rx) ⊃ (∀x)[(Jx • Ix) ⊃ Wx] (∃x)[(Jx • Ix) • ∼Ax]
∼(∃x)(Ax • Wx) / (∀x)(Jx ⊃ Rx)
E) (∃x)(Jx • ∼Rx) ⊃ (∀x)[(Jx • Ix) ⊃ Wx] (∃x)[(Jx • Ix) • Ax]
∼(∃x)∼(Ax • Wx) / (∀x)(Jx ⊃ Rx)

Accepted Answer

The correct translation reflects the structure of the argument: some journalists are not respectable implies all journalists on the web have work to do; some journalists on the web lack assignments; it is not the case that something without an assignment has work to do; therefore, all journalists are respectable. Option B accurately captures these premises and the conclusion using logical notation.

Question 13

If some journalists are not respectable, then all journalists on the web have work to do. Some journalists on the web lack assignments. It is not the case that something without an assignment has work to do. So all journalists are respectable.
-Which of the following propositions is an immediate (one-step) consequence in M of the given premises?
A) (Jx • Ix) • ∼Ax
B) (∀x)∼(∼Ax • Wx)
C) ∼Aa • Wa
D) (Ja • ∼Ra) ⊃ (∀x)[(Jx • Ix) ⊃ Wx]
E) (∃x)(Jx • ∼Rx) ⊃ ∼(∀x)∼[(Jx • Ix) ⊃ Wx]

Accepted Answer

The given premises can be symbolized as:

1. Some journalists are not respectable: ∃x(Jx ∧ ¬Rx)
2. If some journalists are not respectable, then all journalists on the web have work to do: ∀x[(Jx ∧ ¬Rx) → Wx]
3. Some journalists on the web lack assignments: ∃x(Jx ∧ Wx ∧ ¬Ax)
4. It is not the case that something without an assignment has work to do: ∀x[¬(Wx ∧ ¬Ax) → ¬Wx]
Conclusion: All journalists are respectable: ∀x(Jx → Rx)

By combining premises 1 and 2, we can derive ¬(∀x(Jx → Rx)). This can be simplified to (∃xJx) ∧ (∃x¬Rx), which also follows from premise 3. From here, we can apply De Morgan's law to the second part of the derived sentence to get ∃x(Jx ∧ ¬Rx). But this is just what our first premise stated. Therefore, it is not possible to derive any new conclusion from the given premises using one step of deduction. However, by examining the form of the answer choices, we can see that only option B is a universally quantified statement that involves the negation of an antecedent and the denial of a consequent, which matches a common form of the contrapositive. We can show that it is true by contraposition:

¬(∀x)[¬(¬Ax ∧ Wx)]
∃x ¬[¬Ax ∧ Wx]
∃x (Ax ∨ ¬Wx)
∀x ¬(Ax ∨ ¬Wx)
∀x (¬Ax ∧ Wx)

The last statement means that anyone without an assignment is a journalist on the web, which is true given that our premises already imply that some journalists on the web lack assignments. Therefore, the best choice is B.

Question 14

If some journalists are not respectable, then all journalists on the web have work to do. Some journalists on the web lack assignments. It is not the case that something without an assignment has work to do. So all journalists are respectable.
-Which of the following claims can also be derived from the premises of this argument?
A) (∃x)(Jx • ∼Ix)
B) (∃x)(Wx $ \lor $ Ax)
C) (∀x)[∼(Rx $ \lor $ Px) ⊃ Rx]
D) (∃x)[∼Ix • ∼(Rx $ \lor $ Px)]
E) (∀x)[∼(Rx • Px) ⊃ ∼Rx]

Accepted Answer

The answer of If some journalists are not respectable, then...

Question 15

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

Accepted Answer

The answer of 1. (∀x)(Jx ⊃ Kx)
2. (∀x)(Jx ⊃ ∼Lx)
-Consider...

Question 16

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

Accepted Answer

From the premises, it's clear that for all x, if Jx then Kx, and if Jx then not Lx. This directly leads to the conclusion that for all x, if Jx, then both Kx and not Lx, which is what option E states.

Question 17

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

Accepted Answer

To prove the conclusion (∀x)∼Jx indirectly, we assume the negation of the conclusion, which is ∼(∀x)∼Jx. This assumption allows us to derive a contradiction, thereby supporting the original conclusion through reductio ad absurdum.

Question 18

1. (∀x)Ix $ \lor $ ∼(∃x)Hx
2. (∃x)Jx ⊃ ∼(∀x)Ix
3. Hc / (∀x)∼Jx
-Which of the following propositions is not a likely last line of the indented sequence for an indirect proof of the given argument?
A) Hc • ∼Hc
B) (∃x)Hx • ∼(∃x)Hx
C) Id • ∼Id
D) (∃x)Jx • ∼(∃x)Jx
E) (∀x)Ix • ∼(∀x)Ix

Accepted Answer

Assuming ∼(∀x)Ix and Hc, we can use Universal Instantiation to get ∼Ic. Using Modus Tollens with the first premise, we can conclude (∃x)Hx. Then, assuming (∃x)Hx, we can use Existential Instantiation to get Hd. Using Modus Tollens with the second premise, we can conclude ∼(∀x)Ix. Combining this with our previous assumption, we get a contradiction of ∼Ic and Ic. Therefore, we can conclude (∀x)∼Jx.

Option B, (∃x)Hx • ∼(∃x)Hx, is not a likely last line of the indented sequence for an indirect proof because it does not follow directly from the steps taken in the proof, and it is not a contradiction that would result in a proof by contradiction.

Question 19

1. (∀x)(Px ⊃ Qx)
2. ∼(∃x)[(Px • Rx) • Qx]
3. (∃x)Rx / ∼(∀x)Px
-Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?
A) ∼Px
B) ∼(∀x)∼Px
C) Px
D) (∀x)Px
E) ∼(∀x)Px

Accepted Answer

To prove ∼(∀x)Px indirectly, assuming the opposite of what you want to prove is a standard approach. Here, assuming (∀x)Px (D) is the opposite of the conclusion ∼(∀x)Px, making it the appropriate assumption for an indirect proof.

Question 20

1. (∀x)(Px ⊃ Qx)
2. ∼(∃x)[(Px • Rx) • Qx]
3. (∃x)Rx / ∼(∀x)Px
-Which of the following propositions isnot a likely last line of the indented sequence for an indirect proof of the given argument?
A) Rd • ∼Rd
B) (∃x)(Px • Rx) • ∼(∃x)(Px • Rx)
C) Qd • ∼Qd
D) (∀x)(Px • Rx) • ∼(∀x)(Px • Rx)
E) Pd • ∼Pd

Accepted Answer

Option B is not a likely last line of an indirect proof for the given argument because it directly contradicts the premise 2, which states that there does not exist an x such that Px and Rx and Qx are all true. An indirect proof typically aims to show a contradiction arises from assuming the negation of what you're trying to prove, but option B does not align with proving the conclusion ∼(∀x)Px from the given premises.

Quiz 4: Monadic Predicate Logic

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

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

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

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

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

1. ∼(∃x) [Fx • (Gx • Hx) ] 2. ∼(∃x) (Ix • ∼Fx) -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?

If anything is stuck in traffic, Bruno will be mad if it is late. Something is neither annoyed nor not late. If Bruno is mad, then something is annoyed. So something is not stuck in traffic. -Which of the following is the best translation into M of this argument?

If anything is stuck in traffic, Bruno will be mad if it is late. Something is neither annoyed nor not late. If Bruno is mad, then something is annoyed. So something is not stuck in traffic. -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

If anything is stuck in traffic, Bruno will be mad if it is late. Something is neither annoyed nor not late. If Bruno is mad, then something is annoyed. So something is not stuck in traffic. -Which of the following claims can also be derived from the premises of this argument?

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?

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

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?

1. (∀x) Ix $\lor$ ∼(∃x) Hx 2. (∃x) Jx ⊃ ∼(∀x) Ix 3. Hc / (∀x) ∼Jx -Which of the following propositions is not a likely last line of the indented sequence for an indirect proof of the given argument?

1. (∀x) (Px ⊃ Qx) 2. ∼(∃x) [(Px • Rx) • Qx] 3. (∃x) Rx / ∼(∀x) Px -Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

1. (∀x) (Px ⊃ Qx) 2. ∼(∃x) [(Px • Rx) • Qx] 3. (∃x) Rx / ∼(∀x) Px -Which of the following propositions isnot a likely last line of the indented sequence for an indirect proof of the given argument?