Question 1

1. ∼A ≡ B
2. A • ∼C
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) ∼(A ≡ B)
B) B ≡ ∼A
C) ∼∼A ≡ ∼B
D) Options A, B, and C
E) Options A and B, but not option C

Accepted Answer

The given premises are:1. ∼A ≡ B2. A • ∼CFrom premise 1, we can derive:∼(A ≡ B) ≡ ∼BFrom premise 2, we can derive:A ≡ ∼∼CSubstituting the derived equivalence from premise 2 into the derived equivalence from premise 1, we get:∼(A ≡ B) ≡ ∼∼CSimplifying this equivalence, we get:∼∼A ≡ ∼BTherefore, options A, B, and C are all immediate consequences of the given premises.

Question 2

1. ∼A ≡ B
2. A • ∼C
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?
A) ∼(B $ \lor $ C)
B) ∼(A $ \lor $ C)
C) ∼(A $ \lor $ B)
D) ∼(∼B $ \lor $ C)
E) ∼(A $ \lor $ ∼C)

Accepted Answer

The answer of 1. ∼A ≡ B
2. A • ∼C
-Which...

Question 3

1. D • ∼E
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premise?
A) D
B) ∼E
C) ∼(D • ∼E)
D) E • ∼D
E) None of the above

Accepted Answer

The answer of 1. D • ∼E
-Which of the following...

Question 4

1. D • ∼E
-Which of the following propositions is derivable from the given premise using any of the twenty-five rules of PL?
A) D ≡ E
B) ∼D ≡ E
C) ∼D ≡ ∼E
D) D ⊃ E
E) ∼D ⊃ ∼E

Accepted Answer

The answer of 1. D • ∼E
-Which of the following...

Question 5

1. (F $ \lor $ G) ≡ H
2. H • ∼G
3. F ≡ I
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) F ≡ H
B) H ≡ (F $ \lor $ G)
C) ∼F ≡ I
D) I ≡ ∼F
E) ∼(F $ \lor $ G) ≡ H

Accepted Answer

B is correct because it is simply a restatement of the first premise, (F $ \lor $ G) ≡ H, which is a given.

Question 6

1. (F $ \lor $ G) ≡ H
2. H • ∼G
3. F ≡ I
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?
A) ∼F
B) G
C) ∼H
D) I
E) ∼I

Accepted Answer

Given F ≡ I (premise 3), we can directly derive that I is true if F is true, and vice versa, without needing to manipulate the other premises. This is a straightforward application of the equivalence rule.

Question 7

1. (J ≡ K) $ \lor $ L
2. K ≡ L
3. ∼L
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) ∼K ≡ L
B) ∼L ≡ ∼K
C) ∼L ⊃ (J ≡ K)
D) L ≡ K
E) ∼K

Accepted Answer

The answer of 1. (J ≡ K) $ \lor $...

Question 8

1. (J ≡ K) $ \lor $ L
2. K ≡ L
3. ∼L
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?
A) J
B) K
C) L
D) ∼(J $ \lor $ K)
E) ∼(J $ \lor $ ∼L)

Accepted Answer

The answer of 1. (J ≡ K) $ \lor $...

Question 9

1. M ≡ ∼N
2. ∼M ⊃ O
3. ∼N ⊃ O
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) (M ⊃ N) • (N ⊃ M)
B) (M ⊃ ∼N) • (∼N ⊃ M)
C) (M • N) $ \lor $ (M • ∼N)
D) ∼M ≡ ∼N
E) N ≡ ∼M

Accepted Answer

Applying Modus Tollens to premises 2 and 3 yields ∼O ⊃ M and ∼O ⊃ N, respectively. Combining these two statements using conjunction yields (∼O ⊃ M) • (∼O ⊃ N), which is logically equivalent to (M ⊃ ∼N) • (∼N ⊃ M).

Question 10

1. M ≡ ∼N
2. ∼M ⊃ O
3. ∼N ⊃ O
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?
A) M
B) ∼M
C) N
D) O
E) P

Accepted Answer

From premise 1, we know that M is equivalent to the negation of N. From premises 2 and 3, we know that if either M or N are false (negated), then O must be true. Combining these statements, we can conclude that if M is false (i.e. N is true), then O must be true, since ∼N ≡ M. Similarly, if N is false (i.e. M is true), then O must also be true. Therefore, O is derivable from the given premises.

Question 11

1. (P • ∼Q) ≡ (R $ \lor $ S)
2. ∼(R ≡ ∼T)
3. (T $ \lor $ ∼S) • (T $ \lor $ P)
4. ∼T
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) ∼R ≡ ∼∼T
B) ∼R ≡ T
C) ∼R ≡ ∼T
D) Options A, B, and C
E) Options A and B, but not option C

Accepted Answer

Starting from the first premise, we can use the equivalence between a conditional and its contrapositive, as well as De Morgan's law, to rewrite it as:

(P • Q') \ (R v S')

From the third premise, we know that T v ~S, and from the fourth premise, we know that ~T. Combining these, we have:

~S

Using this with the second premise, we have:

R ≡ T'

Which can be further simplified as:

R ≡ ~T

Substituting this into our rewritten first premise, we get:

(P • Q') \ (R v S')
(P • Q') \ (~T v S')
(P • ~S) \ (~T • ~S')
~(T ≡ ~R)

The last line can be rearranged and simplified to:

~R ≡ ~T

This matches option C. None of the other options are correct.

Question 12

1. (P • ∼Q) ≡ (R $ \lor $ S)
2. ∼(R ≡ ∼T)
3. (T $ \lor $ ∼S) • (T $ \lor $ P)
4. ∼T
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?
A) ∼P
B) Q
C) R
D) S
E) T

Accepted Answer

Given ∼T from premise 4, we can use it in premise 3, (T ∨ ∼S) • (T ∨ P), to deduce that ∼S must be true for the first part of the conjunction to hold, since T is false. With ∼S, looking at premise 1, (P • ∼Q) ≡ (R ∨ S), and knowing S is false, for the equivalence to hold, P • ∼Q must also be false. Since we don't have information directly about P, we focus on the other side of the equivalence, R ∨ S, which now depends solely on R being true or false for the equivalence to hold. However, from premise 2, ∼(R ≡ ∼T), knowing T is false, R cannot be equivalent to true (since ∼T is true), suggesting R has to be true for the statement to be not equivalent (since if R were false, it would be equivalent to ∼T, which is true). However, this does not directly lead to any of the options being derivable without further assumptions. The mistake in the initial explanation is assuming specific truth values for R and the direct derivability of other options without considering all implications of the premises. The correct approach involves analyzing the logical structure and implications of the premises more carefully, particularly how they interact with each other under the rules of propositional logic. Given the premises, without a direct path to deduce the truth value of Q, R, S, or P from the given information and the mistake in the initial reasoning, the correct derivation from the premises is not straightforwardly provided in the explanation. The correct answer relies on a detailed analysis of the premises and the application of propositional logic rules, which was not accurately captured in the initial explanation.

Question 13

1. V ≡ (W $ \lor $ ∼X)
2. ∼Y ≡ ∼V
3. (W ⊃ Y) ⊃ Z
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) ∼(Y ≡ V)
B) ∼(∼Y ≡ ∼V)
C) Y ≡ ∼V
D) V ≡ ∼Y
E) Y ≡ V

Accepted Answer

The answer of 1. V ≡ (W $ \lor $...

Question 14

1. V ≡ (W $ \lor $ ∼X)
2. ∼Y ≡ ∼V
3. (W ⊃ Y) ⊃ Z
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?
A) V
B) Y
C) Z
D) W ⊃ Z
E) V ⊃ W

Accepted Answer

From premise 2, we have ∼V ≡ ∼Y. Substituting ∼V for ∼Y in premise 3 gives (W ⊃ ∼V) ⊃ Z. We can use Material Implication rule to rewrite this as ∼(W ⊃ ∼V) ∨ Z. Now, using De Morgan's law, we get (W ∧ V) ∨ Z. Finally, from premise 1, we have V ≡ (W $ \lor $ ∼X). Substituting V for (W $ \lor $ ∼X) in the previous equation gives (W ∧ (W $ \lor $ ∼X)) ∨ Z. By Distribution rule, we get (W ∨ Z) ∧ ((W $ \lor $ ∼X) ∨ Z). Now, by Commutation rule, we can rewrite this as (W $ \lor $ ∼X) ∨ Z. Therefore, the proposition Z is derivable from the given premises.

Question 15

The dentist is pleased if, and only if, I both brush and floss. The dentist is not pleased, but I brush. So, I don't floss.
-Which of the following is the best translation into PL of this argument?
A) P ⊃ (B • F) ∼P • F / ∼F
B) P ≡ (B • F) ∼P • B / ∼F
C) P ≡ (B • F) ∼P $ \lor $ B / ∼F
D) (P ≡ B) • F ∼P $ \lor $ B / ∼F
E) (P ≡ B) • F ∼P • B / ∼F

Accepted Answer

The correct translation reflects the logical structure of the argument: "The dentist is pleased if and only if I both brush and floss" translates to $P \equiv (B \cdot F)$, indicating that the dentist being pleased is equivalent to both brushing and flossing happening. "The dentist is not pleased, but I brush" translates to $\sim P \cdot B$, indicating the dentist is not pleased and I brush. The conclusion "So, I don't floss" translates to $\sim F$. This structure is accurately represented in choice B.

Question 16

The dentist is pleased if, and only if, I both brush and floss. The dentist is not pleased, but I brush. So, I don't floss.
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) ∼P ≡ (B • F)
B) ∼P ≡ ∼(B • F)
C) ∼[P ≡ (B • F)]
D) ∼(B • F) ≡ P
E) None of the above

Accepted Answer

Let P be "The dentist is pleased"
Let B be "I brush"
Let F be "I floss"
From the premises, we know that P is false (the dentist is not pleased) and B is true (I brush). Therefore, using the fact that the dentist is pleased if and only if I both brush and floss, we can conclude that F is false (I don't floss).
The equivalent form of "I don't floss" in PL is ∼(B • F). Therefore, the immediate consequence in PL of the given premises is ∼P ≡ ∼(B • F), which is choice B.

Question 17

The dentist is pleased if, and only if, I both brush and floss. The dentist is not pleased, but I brush. So, I don't floss.
-Working backward from the conclusion of this argument, which of the following is the most likely justification of the last step of the derivation?
A) MT
B) BMP
C) BMT
D) Simp
E) DS

Accepted Answer

The answer of The dentist is pleased if, and only...

Question 18

I know something if, and only if, I have some causal contact with the physical world. If I know something, then there is no demon deceiver. I know something. So, I know something just in case both I have causal contact with the physical world and there is no demon deceiver.
-Which of the following is the best translation into PL of this argument?
A) K ≡ C K ⊃ ∼D
K / K ≡ (C • ∼D)
B) K ≡ C K ⊃ ∼D
K / (K ≡ C) • ∼D
C) K ≡ C K ≡ ∼D
K / K ≡ (C • ∼D)
D) K ≡ C ∼D ⊃ K
K / K ≡ (C • ∼D)
E) K ≡ C ∼D ⊃ K
K / (K ≡ C) • ∼D

Accepted Answer

The argument structure is best represented by choice A, which correctly translates the given statements into propositional logic, maintaining the logical relationships between knowing something (K), having causal contact with the physical world (C), and the absence of a demon deceiver (∼D).

Question 19

I know something if, and only if, I have some causal contact with the physical world. If I know something, then there is no demon deceiver. I know something. So, I know something just in case both I have causal contact with the physical world and there is no demon deceiver.
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) (K ⊃ C) • (C ⊃ K)
B) K $ \lor $ D
C) K $ \lor $ ∼D
D) All of the above
E) None of the above

Accepted Answer

The correct answer is D because all of the propositions are immediate consequences of the given premises.

Question 20

I know something if, and only if, I have some causal contact with the physical world. If I know something, then there is no demon deceiver. I know something. So, I know something just in case both I have causal contact with the physical world and there is no demon deceiver.
-Working backward from the conclusion of this argument, which of the following is the most likely justification of the last step of the derivation?
A) BMP
B) BMT
C) BAssoc
D) BHS
E) Equiv

Accepted Answer

The last step of the derivation uses equivalence (Equiv.), which means that the conclusion "I know something just in case both I have causal contact with the physical world and there is no demon deceiver" is derived by showing that knowing something is equivalent to having causal contact with the physical world and there being no demon deceiver. This is because the argument establishes a bi-conditional relationship between knowing something and the two conditions (causal contact and no demon deceiver), which is what equivalence involves.

Quiz 3: Inference in Propositional Logic

1. ∼A ≡ B 2. A • ∼C -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. ∼A ≡ B 2. A • ∼C -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?

1. D • ∼E -Which of the following propositions is an immediate (one-step) consequence in PL of the given premise?

1. D • ∼E -Which of the following propositions is derivable from the given premise using any of the twenty-five rules of PL?

1. (F $\lor$ G) ≡ H 2. H • ∼G 3. F ≡ I -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. (F $\lor$ G) ≡ H 2. H • ∼G 3. F ≡ I -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?

1. (J ≡ K) $\lor$ L 2. K ≡ L 3. ∼L -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. (J ≡ K) $\lor$ L 2. K ≡ L 3. ∼L -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?

1. M ≡ ∼N 2. ∼M ⊃ O 3. ∼N ⊃ O -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. M ≡ ∼N 2. ∼M ⊃ O 3. ∼N ⊃ O -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?

1. (P • ∼Q) ≡ (R $\lor$ S) 2. ∼(R ≡ ∼T) 3. (T $\lor$ ∼S) • (T $\lor$ P) 4. ∼T -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. (P • ∼Q) ≡ (R $\lor$ S) 2. ∼(R ≡ ∼T) 3. (T $\lor$ ∼S) • (T $\lor$ P) 4. ∼T -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?

1. V ≡ (W $\lor$ ∼X) 2. ∼Y ≡ ∼V 3. (W ⊃ Y) ⊃ Z -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. V ≡ (W $\lor$ ∼X) 2. ∼Y ≡ ∼V 3. (W ⊃ Y) ⊃ Z -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?

The dentist is pleased if, and only if, I both brush and floss. The dentist is not pleased, but I brush. So, I don't floss. -Which of the following is the best translation into PL of this argument?

The dentist is pleased if, and only if, I both brush and floss. The dentist is not pleased, but I brush. So, I don't floss. -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

The dentist is pleased if, and only if, I both brush and floss. The dentist is not pleased, but I brush. So, I don't floss. -Working backward from the conclusion of this argument, which of the following is the most likely justification of the last step of the derivation?

Quiz 3: Inference in Propositional Logic

1. ∼A ≡ B 2. A • ∼C -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. ∼A ≡ B 2. A • ∼C -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?

1. D • ∼E -Which of the following propositions is an immediate (one-step) consequence in PL of the given premise?

1. D • ∼E -Which of the following propositions is derivable from the given premise using any of the twenty-five rules of PL?

1. (F ∨ \lor ∨ G) ≡ H 2. H • ∼G 3. F ≡ I -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. (F ∨ \lor ∨ G) ≡ H 2. H • ∼G 3. F ≡ I -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?

1. (J ≡ K) ∨ \lor ∨ L 2. K ≡ L 3. ∼L -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. (J ≡ K) ∨ \lor ∨ L 2. K ≡ L 3. ∼L -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?