Question 1

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

Accepted Answer

The answer of 1. ∼E ⊃ F
2. G • ∼E
-Which...

Question 2

1. ∼E ⊃ F
2. G • ∼E
-Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN)?
A) E
B) F
C) ∼G
D) ∼E ⊃ ∼F
E) E $ \lor $ ∼F

Accepted Answer

Given the premises, we can derive F using Modus Ponens (MP). From the first premise, if not E (∼E) then F (∼E ⊃ F), and the second premise states G and not E (G • ∼E). By simplification (Simp), we can derive ∼E from the second premise, and then by applying MP with the first premise, we can derive F.

Question 3

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

Accepted Answer

The answer of 1. (C $ \lor $ E) \(...

Question 4

1. (C $ \lor $ E) $ \lor $ D
2. ∼E
-Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN)?
A) C $ \lor $ D
B) ∼C $ \lor $ D
C) C
D) D
E) ∼C

Accepted Answer

Given the premises (C $ \lor $ E) $ \lor $ D and ∼E, we can derive C $ \lor $ D using Disjunctive Syllogism (DS) by eliminating E and its negation (∼E), leaving us with C $ \lor $ D.

Question 5

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

Accepted Answer

The answer of 1. N ⊃ ∼O
2. P ⊃ ∼Q
3....

Question 6

1. N ⊃ ∼O
2. P ⊃ ∼Q
3. (N $ \lor $ P) $ \lor $ R
4. ∼R
-Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN)?
A) N • P
B) ∼N • ∼P
C) ∼(N • P)
D) ∼O • ∼Q
E) ∼(O • Q)

Accepted Answer

The answer of 1. N ⊃ ∼O
2. P ⊃ ∼Q
3....

Question 7

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

Accepted Answer

The answer of 1. F ⊃ (C $ \lor $...

Question 8

1. F ⊃ (C $ \lor $ D)
2. ∼[C $ \lor $ (D $ \lor $ E)]
-Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN)?
A) F
B) ∼F
C) C
D) D
E) E

Accepted Answer

From the premises given, we can derive ∼F using De Morgan's laws (DM) and Modus Tollens (MT). The second premise negates the disjunction of C, D, or E. Applying De Morgan's laws to this negation gives us ∼C ∧ ∼(D ∨ E), which further simplifies to ∼C ∧ ∼D ∧ ∼E. Given the first premise, if C or D were true, then F would be true. However, since we have established that both C and D are not true (∼C and ∼D), it follows by Modus Tollens that F cannot be true, hence ∼F is derivable.

Question 9

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

Accepted Answer

The answer of 1. (A $ \lor $ B) ⊃...

Question 10

1. (A $ \lor $ B) ⊃ ∼C
2. C $ \lor $ D
3. E • A
4. F
-Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN)?
A) ∼A
B) C • F
C) C
D) ∼D
E) D • F

Accepted Answer

E) D • F is derivable from the given premises. From premise 2 (C $ \lor $ D) and premise 1 ((A $ \lor $ B) ⊃ ∼C), we can infer D if we assume C leads to a contradiction (since if C were true, it would contradict the first premise given A or B is true). Since we also have F as a given premise (4), we can use Conjunction (Conj) to derive D • F.

Question 11

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

Accepted Answer

C) H • K is an immediate consequence because it directly follows from premise 3 (K • H) by the commutativity of conjunction, which allows us to switch the order of the conjuncts without changing the meaning.

Question 12

1. G ⊃ ∼(H $ \lor $ I)
2. J $ \lor $ G
3. K • H
-Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN)?
A) J • K
B) J • I
C) G • K
D) G ⊃ (H $ \lor $ I)
E) ∼H • ∼I

Accepted Answer

The answer of 1. G ⊃ ∼(H $ \lor $...

Question 13

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

Accepted Answer

E is the correct answer because it is the logical equivalent of the first premise, ∼G • (∼H • ∼I), with the order of the conjunctions rearranged. This rearrangement does not change the logical meaning, making it an immediate consequence.

Question 14

1. ∼G • (∼H • ∼I)
2. (J $ \lor $ H) $ \lor $ K
3. J ⊃ L
4. K ⊃ M
-Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN)?
A) L $ \lor $ M
B) J $ \lor $ H
C) G $ \lor $ I
D) H $ \lor $ I
E) H $ \lor $ K

Accepted Answer

1. Simplification: ∼H • ∼I (from 1)
2. Disjunctive syllogism: J ⊃ L, ∼L ⊢ ∼J (from 3)
3. Disjunctive syllogism: J ⊃ L, K ⊃ M, ∼M ⊢ ∼J (from 4)
4. Addition: ∼H ⊢ ∼H ∨ L (from 1)
5. Modus ponens: J ⊃ L, ∼H ∨ L ⊢ L (from 2 and 4)
6. Modus tollens: ∼J, J ⊃ L ⊢ ∼L (from 3)
7. Modus tollens: ∼J, J ⊃ L, L ⊢ ¬J (from 6) 
8. Disjunctive syllogism: H ∨ K, ∼H ⊢ K (from 2)
9. Modus ponens: K ⊃ M, K ⊢ M (from 4)
10. Addition : M ⊢ L ∨ M 
11. Disjunctive syllogism: H ∨ K, K ⊢ H (from 2)
12. Modus ponens: J ⊃ L, J ⊢ L (from 3)
So from step 5, 10, L ∨ M is established as a theorem, which is equivalent to L \/ M.

Question 15

After-school activities running late entails that the buses will run late. The buses running late is a sufficient condition for either Carlos and Deandra getting home late or Edna missing dinner. Either after-school activities run late but Carlos doesn't get home late, or after-school activities run late but Deandra doesn't get home late. So, Edna misses dinner.
-Which of the following is the best translation into PL of this argument?
A) A ⊃ B B ⊃ [(C • D) $ \lor $ E]
(A $ \lor $ ∼C) $ \lor $ (A $ \lor $ ∼D) / E
B) A ⊃ B B ⊃ [(C • D) $ \lor $ E]
(A • ∼C) $ \lor $ (A • ∼D) / E
C) A ⊃ B [(C • D) $ \lor $ E] ⊃ B
(A • ∼C) $ \lor $ (A • ∼D) / E
D) A ⊃ B [(C • D) $ \lor $ E] ⊃ B
(A $ \lor $ ∼C) $ \lor $ (A $ \lor $ ∼D) / E
E) A ⊃ B [(C • D) $ \lor $ E] ⊃ B
(A $ \lor $ ∼C) • (A $ \lor $ ∼D) / E

Accepted Answer

Option B correctly translates the argument into propositional logic (PL). It captures the structure of the argument by stating that after-school activities running late (A) is a sufficient condition for the buses running late (B), which in turn is a sufficient condition for either both Carlos and Deandra getting home late (C • D) or Edna missing dinner (E). The premises that either after-school activities run late but Carlos doesn't get home late, or after-school activities run late but Deandra doesn't get home late, are correctly represented as a disjunction of conjunctions (A • ∼C) $ \lor $ (A • ∼D), leading to the conclusion that Edna misses dinner (E).

Question 16

After-school activities running late entails that the buses will run late. The buses running late is a sufficient condition for either Carlos and Deandra getting home late or Edna missing dinner. Either after-school activities run late but Carlos doesn't get home late, or after-school activities run late but Deandra doesn't get home late. So, Edna misses dinner.
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) A • (∼C $ \lor $ ∼D)
B) A • (∼C • ∼D)
C) A $ \lor $ (∼C $ \lor $ ∼D)
D) A $ \lor $ (∼C • ∼D)
E) A ⊃ [(C • D) • E]

Accepted Answer

After-school activities running late entails that the buses will run late. The buses running late is a sufficient condition for either Carlos and Deandra getting home late or Edna missing dinner. Either after-school activities run late but Carlos doesn't get home late, or after-school activities run late but Deandra doesn't get home late. So, Edna misses dinner.The correct answer is A) A • (∼C $ \lor $ ∼D) because it is the only proposition that is an immediate (one-step) consequence in PL of the given premises.

Question 17

After-school activities running late entails that the buses will run late. The buses running late is a sufficient condition for either Carlos and Deandra getting home late or Edna missing dinner. Either after-school activities run late but Carlos doesn't get home late, or after-school activities run late but Deandra doesn't get home late. So, Edna misses dinner.
-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) Simp
C) DN
D) DS
E) HS

Accepted Answer

The last step of the argument is a disjunctive syllogism (DS), which concludes that Edna misses dinner. This is because the given premises establish that if the buses run late, either Carlos and Deandra will get home late or Edna will miss dinner. The argument then presents two cases where the buses run late but Carlos or Deandra do not get home late, leaving only the possibility that Edna misses dinner.

Question 18

If moral theories measure outcomes or consequences, then they do not concern duty. If trolley cases are compelling, then moral theories measure consequences. Moral theories concern duty. So, trolley cases are not compelling.
-Which of the following is the best translation into PL of this argument?
A) O ⊃ ∼D T ⊃ C
D / ∼T
B) O ⊃ ∼D C ⊃ T
D / ∼T
C) (O • C) ⊃ ∼D T ⊃ C
D / ∼T
D) (O $ \lor $ C) ⊃ ∼D T ⊃ C
D / ∼T
E) (O $ \lor $ C) ⊃ ∼D T ⊃ O
D / ∼T

Accepted Answer

O represents "moral theories measure outcomes or consequences"
D represents "moral theories concern duty"
T represents "trolley cases are compelling"
C represents "moral theories measure consequences"

The argument can be symbolized as: 
(O v C) -> ~D, T -> C, D / ~T

In this way, we can see that choice D is the best representation of the argument as it correctly symbolizes the premises and conclusion of the argument.

Question 19

If moral theories measure outcomes or consequences, then they do not concern duty. If trolley cases are compelling, then moral theories measure consequences. Moral theories concern duty. So, trolley cases are not compelling.
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?
A) ∼(O $ \lor $ C)
B) C
C) ∼∼D
D) ∼∼T
E) ∼T

Accepted Answer

The answer of If moral theories measure outcomes or consequences,...

Question 20

If moral theories measure outcomes or consequences, then they do not concern duty. If trolley cases are compelling, then moral theories measure consequences. Moral theories concern duty. So, trolley cases are not compelling.
-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) Com
B) DS
C) MT
D) MP
E) CD

Accepted Answer

The last step of the argument, "So, trolley cases are not compelling," can be justified by the use of modus tollens (MT). MT is a valid form of deductive reasoning that infers the negation of the consequent from the negation of the antecedent. In this case, the antecedent is "If trolley cases are compelling, then moral theories measure consequences," and its negation is "Trolley cases are not compelling." The consequent is "moral theories measure outcomes or consequences," and its negation is "moral theories do not measure outcomes or consequences." Therefore, by the MT rule, we can conclude that "moral theories do not concern duty."

Quiz 3: Inference in Propositional Logic

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

1. ∼E ⊃ F 2. G • ∼E -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?

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

1. (C $\lor$ E) $\lor$ D 2. ∼E -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?

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

1. N ⊃ ∼O 2. P ⊃ ∼Q 3. (N $\lor$ P) $\lor$ R 4. ∼R -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?

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

1. F ⊃ (C $\lor$ D) 2. ∼[C $\lor$ (D $\lor$ E) ] -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?

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

1. (A $\lor$ B) ⊃ ∼C 2. C $\lor$ D 3. E • A 4. F -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?

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

1. G ⊃ ∼(H $\lor$ I) 2. J $\lor$ G 3. K • H -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?

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

1. ∼G • (∼H • ∼I) 2. (J $\lor$ H) $\lor$ K 3. J ⊃ L 4. K ⊃ M -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?

If moral theories measure outcomes or consequences, then they do not concern duty. If trolley cases are compelling, then moral theories measure consequences. Moral theories concern duty. So, trolley cases are not compelling. -Which of the following is the best translation into PL of this argument?

Quiz 3: Inference in Propositional Logic

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

1. ∼E ⊃ F 2. G • ∼E -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?

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

1. (C ∨ \lor ∨ E) ∨ \lor ∨ D 2. ∼E -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?

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

1. N ⊃ ∼O 2. P ⊃ ∼Q 3. (N ∨ \lor ∨ P) ∨ \lor ∨ R 4. ∼R -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?

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

1. F ⊃ (C ∨ \lor ∨ D) 2. ∼[C ∨ \lor ∨ (D ∨ \lor ∨ E) ] -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?

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

1. (A ∨ \lor ∨ B) ⊃ ∼C 2. C ∨ \lor ∨ D 3. E • A 4. F -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?

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

1. G ⊃ ∼(H ∨ \lor ∨ I) 2. J ∨ \lor ∨ G 3. K • H -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?

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

1. ∼G • (∼H • ∼I) 2. (J ∨ \lor ∨ H) ∨ \lor ∨ K 3. J ⊃ L 4. K ⊃ M -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?