Question 1

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

Accepted Answer

The answer of 1. (F $ \lor $ G) ⊃...

Question 2

1. (F $ \lor $ G) ⊃ H
2. F • E
-Which of the following propositions is not derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD)?
A) F
B) F $ \lor $ G
C) F $ \lor $ ∼G
D) ∼G
E) H

Accepted Answer

The answer of 1. (F $ \lor $ G) ⊃...

Question 3

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

Accepted Answer

From premise 1, ∼W • ∼X, we can immediately conclude ∼W as it is a conjunct in the conjunction.

Question 4

1. ∼W • ∼X
2. Y ⊃ W
3. ∼Y ⊃ Z
-Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD)?
A) Z $ \lor $ X
B) Z • X
C) Y • ∼X
D) Y $ \lor $ ∼X
E) ∼Z

Accepted Answer

From the premises, we can derive Z $ \lor $ X using Disjunctive Syllogism (DS) and Constructive Dilemma (CD). Here's a brief explanation: 1. From premise 1 (∼W • ∼X), we can use Simplification (Simp) to get ∼W.2. From premise 2 (Y ⊃ W) and the derived ∼W, we can use Modus Tollens (MT) to derive ∼Y.3. From premise 3 (∼Y ⊃ Z) and the derived ∼Y, we can use Modus Ponens (MP) to derive Z.4. Since we have Z, we can use Addition (Add) to add any statement to it, thus deriving Z $ \lor $ X.

Question 5

1. J ⊃ (K ⊃ L)
2. M ⊃ K
3. M ⊃ J
4. M • K
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) J
B) M ⊃ L
C) M ⊃ (K ⊃ L)
D) All of the above
E) Options B and C, but not option A

Accepted Answer

The answer of 1. J ⊃ (K ⊃ L)
2. M...

Question 6

1. J ⊃ (K ⊃ L)
2. M ⊃ K
3. M ⊃ J
4. M • K
-Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD)?
A) J
B) K
C) L
D) All of the above
E) Options A and B, but not option C

Accepted Answer

Given the premises and using the rules of inference, we can derive all of the options (J, K, L). Starting with premise 4 (M • K), we can use Simplification (Simp) to derive K directly. Using Modus Ponens (MP) with premises 2 (M ⊃ K) and 3 (M ⊃ J), we can derive J since M implies both K and J. Finally, using Hypothetical Syllogism (HS) with premises 1 (J ⊃ (K ⊃ L)) and our derived J, we can conclude K ⊃ L, and then using MP with our derived K, we can conclude L. Therefore, all of the options A, B, and C (J, K, L) are derivable, making D the correct answer.

Question 7

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

Accepted Answer

From premise 2, we can use double negation elimination to get ∼Q, which combined with premise 1 gives us P ⊃ Q. From there, we can use modus tollens with premise 4 to get ∼P. Finally, we can use modus ponens with P ⊃ Q and ∼P to get ∼Q, which combined with premise 2 gives us R. Therefore, ∼R is an immediate consequence.

Question 8

1. P ⊃ ∼Q
2. R $ \lor $ ∼∼Q
3. R ⊃ S
4. ∼S
-Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD)?
A) (P • Q) • ∼R
B) (∼P • ∼∼Q) • ∼R
C) (∼P • ∼∼S) • ∼R
D) (P • ∼Q) • ∼R
E) (∼P • Q) • R

Accepted Answer

The answer of 1. P ⊃ ∼Q
2. R \( \lor...

Question 9

1. G ⊃ H
2. ∼G ⊃ (I ⊃ J)
3. K ⊃ L
4. I $ \lor $ K
5. ∼H
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) ∼G
B) I ⊃ L
C) H ⊃ (I ⊃ J)
D) All of the above
E) Options A and B, but not option C

Accepted Answer

Since H is false (∼H is given), G must also be false (from 1: G ⊃ H), leading to ∼G.

Question 10

1. G ⊃ H
2. ∼G ⊃ (I ⊃ J)
3. K ⊃ L
4. I $ \lor $ K
5. ∼H
-Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD)?
A) H $ \lor $ L
B) G $ \lor $ L
C) J $ \lor $ L
D) K $ \lor $ J
E) G $ \lor $ K

Accepted Answer

1. G ⊃ H (Given)
2. ∼G ⊃ (I ⊃ J) (Given)
3. K ⊃ L (Given)
4. I $ \lor $ K (Given)
5. ∼H (Given)
From (1) and (5), we can derive ∼G by Modus Tollens (MT). 
From (2) and ∼G, we can derive I ⊃ J by MT. 
From (4) and I, we can derive K by Disjunctive Syllogism (DS). 
From (3) and K, we can derive L by MP. 
Therefore, we can conclude J $ \lor $ L by Addition (Add) and then applying Disjunctive Syllogism (DS) to (2) and J $ \lor $ L to get L.

Question 11

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

Accepted Answer

From premise 4, we have ∼∼F, which simplifies to F. Given premise 3, ∼F $ \lor $ E, and knowing F is true, the only way for the disjunction to hold is if E is true. This is because a disjunction (∼F $ \lor $ E) is true if at least one of its components is true. Since we know ∼F is false (because F is true), E must be true.

Question 12

1. (A ⊃ B) • (C ⊃ D)
2. E ⊃ A
3. ∼F $ \lor $ E
4. ∼∼F
-Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD)?
A) ∼A
B) B
C) ∼B
D) D ⊃ C
E) A ⊃ E

Accepted Answer

From the premises, we can derive B using Modus Ponens (MP) and Hypothetical Syllogism (HS). Here's a brief explanation:1. From (4) ∼∼F, we can derive F using Double Negation (DN).2. From (3) ∼F $ \lor $ E and F, we can derive E using Disjunctive Syllogism (DS).3. From (2) E ⊃ A and E, we can derive A using Modus Ponens (MP).4. From (1) (A ⊃ B) • (C ⊃ D) and A, we can derive B using Modus Ponens (MP) again.

Question 13

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

Accepted Answer

Given that $ \sim A $ is true, from 4, for $ (D \lor E) \supset (A \lor C) $ to hold true, $ D \lor E $ must be false if $ A \lor C $ is false. Since $ A $ is false ($ \sim A $ is given), for $ A \lor C $ to be false, $ C $ must also be false. However, there is no direct information about $ C $ being false. Instead, the immediate consequence from $ \sim A $ and the premises is that $ B \lor C $ is still a possibility without directly negating or affirming $ B $ or $ C $. The premises do not allow for a one-step deduction that negates $ B \lor C $, affirms $ B $, affirms $ C $, or negates $ C $ directly. Therefore, $ B \lor C $ remains a valid proposition as it does not contradict any given premises or the negation of $ A $.

Question 14

1. A $ \lor $ (B $ \lor $ C)
2. B ⊃ D
3. C ⊃ E
4. (D $ \lor $ E) ⊃ (A $ \lor $ C)
5. ∼A
-Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD)?
A) A
B) B
C) C
D) D
E) E

Accepted Answer

Using the premises, we can derive C. From premise 5, we have ∼A. Using premise 1, A $ \lor $ (B $ \lor $ C), and premise 5, we can use Disjunctive Syllogism (DS) to derive B $ \lor $ C. From premise 2, B ⊃ D, and premise 3, C ⊃ E, we can use Constructive Dilemma (CD) with B $ \lor $ C to derive D $ \lor $ E. Using premise 4, (D $ \lor $ E) ⊃ (A $ \lor $ C), and D $ \lor $ E, we can use Modus Ponens (MP) to derive A $ \lor $ C. Since we have ∼A, we can again use DS to derive C.

Question 15

Mica goes swimming only if Nicole lifeguards. Pedro is free on the condition that Ona goes to the beach. Mica goes swimming unless Ona goes to the beach. Nicole doesn't lifeguard. So, either Pedro is free or Mica goes swimming.
-Which of the following is the best translation into PL of this argument?
A) N ⊃ M P ⊃ O
M $ \lor $ O
∼N / P $ \lor $ M
B) N ⊃ M O ⊃ P
M $ \lor $ O
∼N / P $ \lor $ M
C) M ⊃ N P ⊃ O
M $ \lor $ O
∼N / P $ \lor $ M
D) M ⊃ N
O ⊃ P
M $ \lor $ O
∼N / P $ \lor $ M
E) M ≡ N P ⊃ O
M $ \lor $ O
∼N / P $ \lor $ M

Accepted Answer

The argument is best translated as: Mica swims only if Nicole lifeguards (M ⊃ N), Pedro is free if Ona goes to the beach (O ⊃ P), Mica swims unless Ona goes to the beach (M ∨ O), Nicole doesn't lifeguard (∼N), therefore either Pedro is free or Mica swims (P ∨ M).

Question 16

Mica goes swimming only if Nicole lifeguards. Pedro is free on the condition that Ona goes to the beach. Mica goes swimming unless Ona goes to the beach. Nicole doesn't lifeguard. So, either Pedro is free or Mica goes swimming.
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) M
B) ∼M
C) N
D) O
E) ∼O

Accepted Answer

Given that Nicole doesn't lifeguard (¬N), and since Mica goes swimming only if Nicole lifeguards (M → N), it follows directly by modus tollens that Mica does not go swimming (¬M).

Question 17

Mica goes swimming only if Nicole lifeguards. Pedro is free on the condition that Ona goes to the beach. Mica goes swimming unless Ona goes to the beach. Nicole doesn't lifeguard. So, either Pedro is free or Mica goes swimming.
-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) Conj
E) Add

Accepted Answer

The last step of the argument involves introducing a new element that wasn't previously discussed, which is characteristic of the Addition rule (Add) in logic. This rule allows for the introduction of a disjunction (an "or" statement) to the argument.

Question 18

If names are purely referential and do not have descriptive content, then 'Fido' means Fido. If names do have descriptive content, then 'Fido' means my idea of Fido. But Fido does not mean my idea of Fido. Names are purely referential. So, 'Fido' means Fido.
-Which of the following is the best translation into PL of this argument?
A) R • (∼D ⊃ F) D ⊃ I
∼I
R / F
B) (R • ∼D) ⊃ F D ⊃ F
∼I
N / F
C) (R • ∼D) ⊃ F D ⊃ I
∼I
R / F
D) (R • ∼D) ⊃ F 
D ⊃ F
∼F
R / F
E) (R • D) ⊃ F D ⊃ I
∼F
R / F

Accepted Answer

The argument concludes that names are purely referential and do not have descriptive content, and thus 'Fido' means Fido. The premises support this conclusion by stating that if names have descriptive content, then 'Fido' would mean my idea of Fido, but it doesn't. This can be represented in symbolic logic as (R • ∼D) ⊃ F for the conditional premise, D ⊃ I for the premise that 'Fido' does not mean my idea of Fido, and ∼I for the premise that 'Fido' does not have descriptive content. From these, the conclusion F can be derived by modus ponens. Choice C represents this argument structure most accurately.

Question 19

If names are purely referential and do not have descriptive content, then 'Fido' means Fido. If names do have descriptive content, then 'Fido' means my idea of Fido. But Fido does not mean my idea of Fido. Names are purely referential. So, 'Fido' means Fido.
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) F
B) ∼F
C) I
D) D
E) ∼D

Accepted Answer

The answer of If names are purely referential and do...

Question 20

If names are purely referential and do not have descriptive content, then 'Fido' means Fido. If names do have descriptive content, then 'Fido' means my idea of Fido. But Fido does not mean my idea of Fido. Names are purely referential. So, 'Fido' means Fido.
-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) DS
D) Add
E) Simp

Accepted Answer

The last step of the derivation uses Modus Ponens (MP), which is a form of argument that goes from "If P then Q" and "P" to conclude "Q". In this case, "If names are purely referential, then 'Fido' means Fido" and "Names are purely referential" to conclude "'Fido' means Fido".

Quiz 3: Inference in Propositional Logic

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

1. (F $\lor$ G) ⊃ H 2. F • E -Which of the following propositions is not derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD) ?

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

1. ∼W • ∼X 2. Y ⊃ W 3. ∼Y ⊃ Z -Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD) ?

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

1. J ⊃ (K ⊃ L) 2. M ⊃ K 3. M ⊃ J 4. M • K -Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD) ?

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

1. P ⊃ ∼Q 2. R $\lor$ ∼∼Q 3. R ⊃ S 4. ∼S -Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD) ?

1. G ⊃ H 2. ∼G ⊃ (I ⊃ J) 3. K ⊃ L 4. I $\lor$ K 5. ∼H -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. G ⊃ H 2. ∼G ⊃ (I ⊃ J) 3. K ⊃ L 4. I $\lor$ K 5. ∼H -Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD) ?

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

1. (A ⊃ B) • (C ⊃ D) 2. E ⊃ A 3. ∼F $\lor$ E 4. ∼∼F -Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD) ?

1. A $\lor$ (B $\lor$ C) 2. B ⊃ D 3. C ⊃ E 4. (D $\lor$ E) ⊃ (A $\lor$ C) 5. ∼A -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. A $\lor$ (B $\lor$ C) 2. B ⊃ D 3. C ⊃ E 4. (D $\lor$ E) ⊃ (A $\lor$ C) 5. ∼A -Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD) ?

Quiz 3: Inference in Propositional Logic

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

1. (F ∨ \lor ∨ G) ⊃ H 2. F • E -Which of the following propositions is not derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD) ?

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

1. ∼W • ∼X 2. Y ⊃ W 3. ∼Y ⊃ Z -Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD) ?

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

1. J ⊃ (K ⊃ L) 2. M ⊃ K 3. M ⊃ J 4. M • K -Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD) ?

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

1. P ⊃ ∼Q 2. R ∨ \lor ∨ ∼∼Q 3. R ⊃ S 4. ∼S -Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD) ?

1. G ⊃ H 2. ∼G ⊃ (I ⊃ J) 3. K ⊃ L 4. I ∨ \lor ∨ K 5. ∼H -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. G ⊃ H 2. ∼G ⊃ (I ⊃ J) 3. K ⊃ L 4. I ∨ \lor ∨ K 5. ∼H -Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD) ?

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

1. (A ⊃ B) • (C ⊃ D) 2. E ⊃ A 3. ∼F ∨ \lor ∨ E 4. ∼∼F -Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD) ?

1. A ∨ \lor ∨ (B ∨ \lor ∨ C) 2. B ⊃ D 3. C ⊃ E 4. (D ∨ \lor ∨ E) ⊃ (A ∨ \lor ∨ C) 5. ∼A -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. A ∨ \lor ∨ (B ∨ \lor ∨ C) 2. B ⊃ D 3. C ⊃ E 4. (D ∨ \lor ∨ E) ⊃ (A ∨ \lor ∨ C) 5. ∼A -Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD) ?