Question 1

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

Accepted Answer

From premise 1, if A implies B, then if B is false (as given in premise 2), A must also be false. Therefore, we can immediately conclude that ¬A is true.

Question 2

1. A ⊃ B
2. ∼B
3. ∼A ⊃ ∼C
-Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS)?
A) A
B) B
C) C
D) ∼C

Accepted Answer

From premise 1, we can use Modus Tollens (MT) with premise 2 to get ∼A. Then, we can use Modus Tollens (MT) again with premise 3 and ∼A to get ∼C. Therefore, ∼C is derivable from the given premises using the rules of section 3.1.

Question 3

1. T ⊃ S
2. S ⊃ R
3. T
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) R
B) S
C) ∼S
D) ∼T

Accepted Answer

Given T (premise 3) and T ⊃ S (premise 1), we can directly infer S through modus ponens.

Question 4

1. T ⊃ S
2. S ⊃ R
3. T
-Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS)?
A) R
B) ∼R
C) ∼S
D) ∼T

Accepted Answer

From premise 1 and premise 3, we can use Modus Ponens (MP) to conclude S. Then, from premise 2 and the derived S, we can again use MP to conclude R. Therefore, proposition A, R, is derivable from the given premises using the rules of section 3.1.

Question 5

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

Accepted Answer

Given ∼P, from 1 we can directly infer Q ⊃ R because P $ \lor $ (Q ⊃ R) with ∼P leaves Q ⊃ R as the only possibility.

Question 6

1. P $ \lor $ (Q ⊃ R)
2. R ⊃ (S • T)
3. ∼P
-Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS)?
A) Q
B) R
C) S
D) T
E) Q ⊃ (S • T)

Accepted Answer

From premise 1, we can use DS (disjunctive syllogism) to get Q ⊃ R. From premise 2, we can use MP (modus ponens) with R ⊃ (S • T) to get S • T. In combination with Q ⊃ R from the previous step, we can use MP again to get Q ⊃ (S • T). Finally, from premise 3, we have ∼P, which means we can use MT (modus tollens) with Q ⊃ R to get ∼Q. Therefore, we can combine ∼Q and Q ⊃ (S • T) to get Q ⊃ (S • T) as the derivable proposition.

Question 7

1. P ⊃ [Q ⊃ (P ⊃ R)]
2. P
3. Q
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) P ⊃ R
B) P ⊃ Q
C) Q ⊃ (P ⊃ Q)
D) Q ⊃ (P ⊃ R)
E) None of the above

Accepted Answer

From premise 1, we can use hypothetical syllogism with premise 3 to get Q ⊃ (P ⊃ R).

Question 8

1. P ⊃ [Q ⊃ (P ⊃ R)]
2. P
3. Q
-Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS)?
A) ∼P
B) R ⊃ P
C) R
D) ∼R
E) Q ⊃ R

Accepted Answer

Using modus ponens with premises 1 and 2, we can derive P ⊃ R. Then, using modus ponens again with premises 3 and P ⊃ R, we can derive R. Therefore, option C is the correct choice.

Question 9

1. (P $ \lor $ ∼R) ⊃ (P ⊃ Q)
2. P $ \lor $ ∼R
3. P
4. Q ⊃ ∼S
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) R
B) ∼R
C) P ⊃ Q
D) Q
E) ∼S

Accepted Answer

From premise 1, we can use modus ponens with premise 3 to get P ⊃ Q as our immediate consequence.

Question 10

1. (P $ \lor $ ∼R) ⊃ (P ⊃ Q)
2. P $ \lor $ ∼R
3. P
4. Q ⊃ ∼S
-Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS)?
A) ∼R
B) P ⊃ R
C) ∼Q
D) ∼S
E) ∼P

Accepted Answer

We can use Modus Ponens (MP) on premise 1 and premise 3 to get (P ⊃ Q). Then, we can use Modus Tollens (MT) on premise 4 and the previously derived (P ⊃ Q) to get ∼S. Therefore, the correct answer is D.

Question 11

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

Accepted Answer

From premises 3 and 4, we can infer ∼N by Modus Tollens. From premise 1, we can then infer P • ∼R by Disjunctive Syllogism, but we cannot immediately infer N, Q, or ∼Q without further derivation.

Question 12

1. N $ \lor $ (P • ∼R)
2. (P • ∼R) ⊃ Q
3. N ⊃ O
4. ∼O
-Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS)?
A) N
B) ∼(P • ∼R)
C) Q
D) ∼P • ∼∼R
E) O

Accepted Answer

(P • ∼R) ⊃ Q (2)N (1)(P • ∼R) (1, MP)Q (2, 3, MP)

Question 13

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

Accepted Answer

From premise 1, we know that if ~J is true, then K must be true. From premise 2, if K is true, then L ⊃ M must be true. From premise 3, if J is true, then M must be true. However, from premise 4, we know that M is false. So, we can conclude that ~J must be true, as it is the only remaining option. None of the other answer choices can be immediately inferred from the given premises.

Question 14

1. ∼J ⊃ K
2. K ⊃ (L ⊃ M)
3. J ⊃ M
4. ∼M
-Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS)?
A) ∼K
B) J
C) ∼M ⊃ L
D) ∼L
E) L

Accepted Answer

Starting with premise 3, J ⊃ M, and using Modus Tollens with premise 4, we can conclude ∼J. Then, using Modus Ponens with premises 1 and ∼J, we can derive ∼K. Finally, using Modus Ponens with premise 2 and ∼K, we can derive ∼(L ⊃ M), which is equivalent to ∼L. Therefore, the correct answer is D, ∼L.

Question 15

Quinn or Raina will be valedictorian. Quinn's being valedictorian entails that she receives an A+ in Spanish. She doesn't receive an A+ in Spanish. So, Raina is valedictorian.
-Which of the following is the best translation into PL of this argument?
A) Q $ \lor $ R Q ⊃ S
∼S / R
B) Q $ \lor $ R S ⊃ Q
∼S / R
C) Q $ \lor $ R Q ≡ S
∼A / R
D) Q $ \lor $ R S ⊃ Q
∼A / R
E) Q $ \lor $ R Q ⊃ S
∼S / Q

Accepted Answer

The argument can be translated into propositional logic as: "Either Quinn (Q) or Raina (R) will be valedictorian. If Quinn is valedictorian, then she receives an A+ in Spanish (Q ⊃ S). She doesn't receive an A+ in Spanish (∼S). Therefore, Raina is valedictorian (R)."

Question 16

Quinn or Raina will be valedictorian. Quinn's being valedictorian entails that she receives an A+ in Spanish. She doesn't receive an A+ in Spanish. So, Raina is valedictorian.
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) Q
B) ∼Q
C) S ⊃ Q
D) ∼(Q ⊃ S)
E) ∼(Q $ \lor $ R)

Accepted Answer

Given that Quinn's being valedictorian entails receiving an A+ in Spanish and she doesn't receive an A+, it directly follows that Quinn cannot be the valedictorian, represented as ∼Q.

Question 17

Quinn or Raina will be valedictorian. Quinn's being valedictorian entails that she receives an A+ in Spanish. She doesn't receive an A+ in Spanish. So, Raina is valedictorian.
-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) MP
B) MT
C) HS
D) DS

Accepted Answer

The argument uses Disjunctive Syllogism (DS), which states that if one of the disjuncts is false, the other must be true. Since Quinn is not valedictorian, Raina must be.

Question 18

Julie is considering either occasionalist or epiphenomenal dualism, or identity theory or eliminative materialism. But she can accept neither occasionalist nor epiphenomenal dualism. And she denies identity theory. So, she favors eliminative materialism.
-Which of the following is the best translation into PL of this argument?
A) (O • E) $ \lor $ (I • M)
∼(O • E)
∼I / M
B) (O • E) $ \lor $ (I • M)
∼(O $ \lor $ E)
∼I / M
C) (O $ \lor $ E) $ \lor $ (I $ \lor $ M)
∼(O $ \lor $ E)
∼I / M
D) (O $ \lor $ E) $ \lor $ (I $ \lor $ M)
∼(O • E)
∼I / M
E) (O $ \lor $ E) • (I $ \lor $ M)
∼(O $ \lor $ E)
∼I / M

Accepted Answer

Option C correctly represents the logical structure of the argument. Julie is considering four options, grouped into two pairs: occasionalist or epiphenomenal dualism, and identity theory or eliminative materialism. She rejects both forms of dualism and identity theory, leaving eliminative materialism as her favored position. The use of $ \lor $ (logical OR) correctly captures the structure of her considerations and rejections.

Question 19

Julie is considering either occasionalist or epiphenomenal dualism, or identity theory or eliminative materialism. But she can accept neither occasionalist nor epiphenomenal dualism. And she denies identity theory. So, she favors eliminative materialism.
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) I
B) O $ \lor $ E
C) I $ \lor $ M
D) ∼(I $ \lor $ M)
E) ∼M

Accepted Answer

The given premises are:1. Julie is considering either occasionalist or epiphenomenal dualism, or identity theory or eliminative materialism.2. She can accept neither occasionalist nor epiphenomenal dualism.3. She denies identity theory.From these premises, we can conclude that Julie favors eliminative materialism. This is because she has ruled out all other options. Therefore, the correct answer is C) I $ \lor $ M.

Question 20

Julie is considering either occasionalist or epiphenomenal dualism, or identity theory or eliminative materialism. But she can accept neither occasionalist nor epiphenomenal dualism. And she denies identity theory. So, she favors eliminative materialism.
-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) MP
B) MT
C) HS
D) DS

Accepted Answer

The last step of the derivation likely uses Disjunctive Syllogism (DS), which allows one to eliminate possibilities until only one remains. Given that Julie rejects occasionalist and epiphenomenal dualism, and also denies identity theory, the only remaining option she favors is eliminative materialism.

Quiz 3: Inference in Propositional Logic

1. A ⊃ B 2. ∼B 3. ∼A ⊃ ∼C -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. A ⊃ B 2. ∼B 3. ∼A ⊃ ∼C -Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS) ?

1. T ⊃ S 2. S ⊃ R 3. T -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. T ⊃ S 2. S ⊃ R 3. T -Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS) ?

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

1. P $\lor$ (Q ⊃ R) 2. R ⊃ (S • T) 3. ∼P -Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS) ?

1. P ⊃ [Q ⊃ (P ⊃ R) ] 2. P 3. Q -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. P ⊃ [Q ⊃ (P ⊃ R) ] 2. P 3. Q -Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS) ?

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

1. (P $\lor$ ∼R) ⊃ (P ⊃ Q) 2. P $\lor$ ∼R 3. P 4. Q ⊃ ∼S -Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS) ?

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

1. N $\lor$ (P • ∼R) 2. (P • ∼R) ⊃ Q 3. N ⊃ O 4. ∼O -Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS) ?

1. ∼J ⊃ K 2. K ⊃ (L ⊃ M) 3. J ⊃ M 4. ∼M -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. ∼J ⊃ K 2. K ⊃ (L ⊃ M) 3. J ⊃ M 4. ∼M -Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS) ?

Quinn or Raina will be valedictorian. Quinn's being valedictorian entails that she receives an A+ in Spanish. She doesn't receive an A+ in Spanish. So, Raina is valedictorian. -Which of the following is the best translation into PL of this argument?

Quinn or Raina will be valedictorian. Quinn's being valedictorian entails that she receives an A+ in Spanish. She doesn't receive an A+ in Spanish. So, Raina is valedictorian. -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?