Question 1

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL.
-1. (J ? K) $ \lor $ L
2. K ? L
3. ?L / ?J

Accepted Answer

The answer of derive the conclusions of each of the...

Question 2

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL.
-1. M ≡ ∼N
2. ∼M ⊃ O
3. ∼N ⊃ O / O

Accepted Answer

The answer of derive the conclusions of each of the...

Question 3

derive the conclusions of each of the following arguments 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 / Q

Accepted Answer

The answer of derive the conclusions of each of the...

Question 4

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL.
-1. V ≡ (W $ \lor $ ∼X)
2. ∼Y ≡ ∼V
3. (W ⊃ Y) ⊃ Z / Z

Accepted Answer

The answer of derive the conclusions of each of the...

Question 5

translate the given paragraphs into arguments written in PL. Then, derive their conclusions 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.

Accepted Answer

The answer of translate the given paragraphs into arguments written...

Question 6

translate the given paragraphs into arguments written in PL. Then, derive their conclusions using any of the twenty-five rules of PL.
-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.

Accepted Answer

The answer of translate the given paragraphs into arguments written...

Question 7

1. G ⊃ H
2. G ⊃ I
-Consider assuming 'G' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?
A) H
B) I
C) H • I
D) Options A, B, and C
E) Options A and B, but not option C

Accepted Answer

Assuming 'G' for conditional proof and given the premises "G ⊃ H" and "G ⊃ I", we can immediately derive both 'H' and 'I' as consequences. However, deriving 'H • I' (H and I) requires an additional step of conjunction, not immediate from the assumption of 'G' alone.

Question 8

1. G ⊃ H
2. G ⊃ I
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?
A) G ≡ H
B) G ≡ (H • I)
C) G ⊃ (H • I)
D) All of the above
E) None of the above

Accepted Answer

From the given premises, we can use the conjunction rule to derive G ⊃ (H • I). However, we cannot derive G ≡ H or G ≡ (H • I) as these require additional information about the relationship between H and I, which is not provided in the premises.

Question 9

1. J ⊃ K
2. ∼J $ \lor $ L
3. (K • L) ≡ M
-Consider assuming 'J' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?
A) K
B) ∼K
C) ∼L
D) L ⊃ J
E) None of the above

Accepted Answer

Assuming J, we can use the first premise to infer K by modus ponens.

Question 10

1. J ⊃ K
2. ∼J $ \lor $ L
3. (K • L) ≡ M
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?
A) J ⊃ ∼K
B) J ⊃ ∼L
C) J ⊃ M
D) M ⊃ J
E) None of the above

Accepted Answer

Starting with premise 1 and 2, we can use Modus Tollens to derive ∼K ∨ ∼L. Then we use De Morgan's law to get ∼(K • L), which is equivalent to ∼M. From premise 3, we know that (K • L) ≡ M. Therefore, if we use a biconditional syllogism, we can infer (K • L) ≡ ∼(K • L), which simplifies to K ≡ ∼L. Finally, we can use Modus Tollens again with premise 1 to derive J ⊃ ∼K, and use the biconditional we just derived to convert this to J ⊃ ∼L. 
Thus, the correct answer is C, J ⊃ M.

Question 11

1. ∼(N $ \lor $ O)
2. P ⊃ ∼Q
-Consider assuming 'N $ \lor $ P' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?
A) ∼N $ \lor $ ∼O
B) ∼N • ∼O
C) ∼N
D) ∼Q ⊃ P
E) O $ \lor $ Q

Accepted Answer

Given the premises and the assumption for conditional proof, the immediate one-step consequence is B) ∼N • ∼O. This follows from the first premise ∼(N $ \lor $ O), which by De Morgan's laws directly translates to ∼N • ∼O.

Question 12

1. ∼(N $ \lor $ O)
2. P ⊃ ∼Q
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?
A) (N $ \lor $ P) ⊃ O
B) (N $ \lor $ P) ⊃ (O • P)
C) (N $ \lor $ P) • (Q $ \lor $ R)
D) (N $ \lor $ P) ⊃ ∼(O $ \lor $ Q)
E) (M $ \lor $ P) ⊃ R

Accepted Answer

Beginning with the first premise, DeMorgan's law can be applied to get ~(N V O) = (~N & ~O). Then, the second premise can be used with modus ponens to get P -> ~Q. We can use disjunctive syllogism with ~Q and ~O to get ~Q V ~O. Then, we can use DeMorgan's law again to get ~(Q & O). Finally, we can use modus tollens with the second premise and ~Q to get ~P. Using ~P and ~O, we can apply disjunctive syllogism to get N. Thus, we have N and ~O, and we can use these two premises to derive D, which is (N V P) -> ~(O V Q).

Question 13

1. ∼R ≡ S
2. T ⊃ R
3. ∼(R $ \lor $ T) ⊃ U
-Consider assuming 'S' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?
A) R ≡ ∼S
B) S ≡ R
C) ∼S ≡ ∼ R
D) S ≡ ∼R
E) S ⊃ ∼R

Accepted Answer

Assuming S for conditional proof, from the first premise we get ∼R. Then from the fourth option, we can see that S ≡ ∼R, which is an immediate consequence since we have already shown ∼R to be true under the assumption of S.

Question 14

1. ∼R ≡ S
2. T ⊃ R
3. ∼(R $ \lor $ T) ⊃ U
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?
A) S ⊃ R
B) S ⊃ T
C) S ⊃ (R • T)
D) S ⊃ U
E) S ⊃ (R • U)

Accepted Answer

From premise 1, we have ∼R ≡ S.

Using Material Implication, we can convert premise 2 to ∼T ∨ R.

By applying DeMorgan's Law on premise 3, we get R • T ⊃ U.

Combining everything together, we can derive:

∼T ∨ (∼R ≡ S) (premise 1)
∼T ∨ (∼S ≡ R) (Material Equivalence on premise 1)
(∼T ∨ ∼S) ≡ R (Commutativity and Associativity of ≡)
R ≡ (∼T ∨ ∼S) (Equivalent Rearrangement)
(R ≡ ∼T) ∨ (R ≡ ∼S) (Material Equivalence)
∼(R • T) ⊃ ∼U (Contrapositive of premise 3)
(R • T) ∨ ∼U (Material Implication)
(R ≡ ∼T) ∨ U (Distributivity of ≡ over ∨)
∼S ∨ U (Substitution of ∼T for R in previous step)
S ⊃ U (Material Implication)

Therefore, choice D is the correct answer.

Question 15

1. A ⊃ (B $ \lor $ C)
2. ∼A $ \lor $ (D $ \lor $ E)
3. ∼(C $ \lor $ E)
-Consider assuming 'A' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?
A) B $ \lor $ C
B) ∼∼A
C) ∼C • ∼E
D) All of the above
E) Options A and B, but not option C

Accepted Answer

With the assumption of 'A' for conditional proof, all the given options become immediate consequences. Option A is directly derived from the first premise by modus ponendo ponens, given the assumption of 'A'. Option B is a direct consequence of assuming 'A', as double negation simply removes the negation. Option C is derived from the third premise using De Morgan's laws.

Question 16

1. A ⊃ (B $ \lor $ C)
2. ∼A $ \lor $ (D $ \lor $ E)
3. ∼(C $ \lor $ E)
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?
A) A ⊃ C
B) A ⊃ E
C) A ⊃ (B • C)
D) A ⊃ (D • E)
E) A ⊃ (B • D)

Accepted Answer

The answer of 1. A ⊃ (B $ \lor $...

Question 17

1. F ⊃ [(G $ \lor $ H) ⊃ I]
-Consider assuming 'F' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?
A) G $ \lor $ H
B) (G $ \lor $ H) ⊃ I
C) [F • (G $ \lor $ H)] ⊃ I
D) All of the above
E) Options B and C, but not option A

Accepted Answer

The answer of 1. F ⊃ [(G $ \lor $...

Question 18

1. F ⊃ [(G $ \lor $ H) ⊃ I]
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?
A) F ⊃ I
B) F ⊃ (G ⊃ I)
C) (F ⊃ G) ⊃ I
D) [F ⊃ (G $ \lor $ H)] ⊃ I
E) I ⊃ F

Accepted Answer

B) F ⊃ (G ⊃ I) is derivable from the given premise by applying the Exportation rule, which allows us to transform F ⊃ [(G $ \lor $ H) ⊃ I] into F & (G $ \lor $ H) ⊃ I, and further into F & G ⊃ I, which is logically equivalent to F ⊃ (G ⊃ I).

Question 19

1. J ⊃ (K ⊃ M)
2. L $ \lor $ ∼M
3. J $ \lor $ (K • ∼L)
-Consider assuming 'J' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?
A) K ⊃ M
B) K
C) M
D) K • ∼L
E) ∼L

Accepted Answer

Given the assumption of 'J' for conditional proof, from the first premise (1. J ⊃ (K ⊃ M)), we can directly infer K ⊃ M as an immediate consequence. This is because the structure of the first premise indicates that if J is true, then the conditional statement (K ⊃ M) must also be true.

Question 20

1. J ⊃ (K ⊃ M)
2. L $ \lor $ ∼M
3. J $ \lor $ (K • ∼L)
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?
A) J ≡ K
B) J ≡ (K ⊃ L)
C) J ≡ L
D) J ≡ M
E) J ≡ (M ⊃ L)

Accepted Answer

- Starting with premise 1 and using conditional elimination (→E), we get K ⊃ M
- From premise 2, we can use De Morgan's law and simplify: ∼(L ∧ M)
- Using premise 3 and conjunction elimination (•E), we get K • ∼L
- Combining our two previous deductions using modus ponens (MP), we get K • ∼L ⊃ ∼M 
- Using another application of MP, we can derive K ⊃ ∼M
- Now we can use transitivity of implication (→ trans) with our previous deduction of K ⊃ M, to get K ⊃ J
- From premise 3, using disjunction elimination (∨E) we can get J or K • ∼L
- If we assume K • ∼L, then from our previous deduction of K ⊃ J, we can use modus ponens to get J, which is what we want to prove.
- If we assume J, we can use disjunction introduction (∨I) to get J ∨ (K ⊃ L)
- Finally, assuming J ∨ (K ⊃ L), we can use disjunction elimination twice: once to assume J and derive J ≡ (K ⊃ L) using conditional introduction (→I) and once to assume K ⊃ L and derive J ≡ (K ⊃ L) using conditional introduction again.

Quiz 3: Inference in Propositional Logic

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL. -1. (J ? K) $\lor$ L 2. K ? L 3. ?L / ?J

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL. -1. M ≡ ∼N 2. ∼M ⊃ O 3. ∼N ⊃ O / O

derive the conclusions of each of the following arguments 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 / Q

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL. -1. V ≡ (W $\lor$ ∼X) 2. ∼Y ≡ ∼V 3. (W ⊃ Y) ⊃ Z / Z

translate the given paragraphs into arguments written in PL. Then, derive their conclusions 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.

1. G ⊃ H 2. G ⊃ I -Consider assuming 'G' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. G ⊃ H 2. G ⊃ I -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. J ⊃ K 2. ∼J $\lor$ L 3. (K • L) ≡ M -Consider assuming 'J' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. J ⊃ K 2. ∼J $\lor$ L 3. (K • L) ≡ M -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. ∼(N $\lor$ O) 2. P ⊃ ∼Q -Consider assuming 'N $\lor$ P' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. ∼(N $\lor$ O) 2. P ⊃ ∼Q -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. ∼R ≡ S 2. T ⊃ R 3. ∼(R $\lor$ T) ⊃ U -Consider assuming 'S' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. ∼R ≡ S 2. T ⊃ R 3. ∼(R $\lor$ T) ⊃ U -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. A ⊃ (B $\lor$ C) 2. ∼A $\lor$ (D $\lor$ E) 3. ∼(C $\lor$ E) -Consider assuming 'A' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. A ⊃ (B $\lor$ C) 2. ∼A $\lor$ (D $\lor$ E) 3. ∼(C $\lor$ E) -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. F ⊃ [(G $\lor$ H) ⊃ I] -Consider assuming 'F' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. F ⊃ [(G $\lor$ H) ⊃ I] -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. J ⊃ (K ⊃ M) 2. L $\lor$ ∼M 3. J $\lor$ (K • ∼L) -Consider assuming 'J' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. J ⊃ (K ⊃ M) 2. L $\lor$ ∼M 3. J $\lor$ (K • ∼L) -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

Quiz 3: Inference in Propositional Logic

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL. -1. (J ? K) ∨ \lor ∨ L 2. K ? L 3. ?L / ?J

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL. -1. M ≡ ∼N 2. ∼M ⊃ O 3. ∼N ⊃ O / O

derive the conclusions of each of the following arguments 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 / Q

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL. -1. V ≡ (W ∨ \lor ∨ ∼X) 2. ∼Y ≡ ∼V 3. (W ⊃ Y) ⊃ Z / Z

translate the given paragraphs into arguments written in PL. Then, derive their conclusions 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.

1. G ⊃ H 2. G ⊃ I -Consider assuming 'G' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. G ⊃ H 2. G ⊃ I -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. J ⊃ K 2. ∼J ∨ \lor ∨ L 3. (K • L) ≡ M -Consider assuming 'J' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. J ⊃ K 2. ∼J ∨ \lor ∨ L 3. (K • L) ≡ M -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. ∼(N ∨ \lor ∨ O) 2. P ⊃ ∼Q -Consider assuming 'N ∨ \lor ∨ P' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. ∼(N ∨ \lor ∨ O) 2. P ⊃ ∼Q -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. ∼R ≡ S 2. T ⊃ R 3. ∼(R ∨ \lor ∨ T) ⊃ U -Consider assuming 'S' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. ∼R ≡ S 2. T ⊃ R 3. ∼(R ∨ \lor ∨ T) ⊃ U -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. A ⊃ (B ∨ \lor ∨ C) 2. ∼A ∨ \lor ∨ (D ∨ \lor ∨ E) 3. ∼(C ∨ \lor ∨ E) -Consider assuming 'A' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. A ⊃ (B ∨ \lor ∨ C) 2. ∼A ∨ \lor ∨ (D ∨ \lor ∨ E) 3. ∼(C ∨ \lor ∨ E) -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. F ⊃ [(G ∨ \lor ∨ H) ⊃ I] -Consider assuming 'F' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. F ⊃ [(G ∨ \lor ∨ H) ⊃ I] -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. J ⊃ (K ⊃ M) 2. L ∨ \lor ∨ ∼M 3. J ∨ \lor ∨ (K • ∼L) -Consider assuming 'J' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. J ⊃ (K ⊃ M) 2. L ∨ \lor ∨ ∼M 3. J ∨ \lor ∨ (K • ∼L) -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL. -1. (J ? K) $\lor$ L 2. K ? L 3. ?L / ?J

derive the conclusions of each of the following arguments 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 / Q

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL. -1. V ≡ (W $\lor$ ∼X) 2. ∼Y ≡ ∼V 3. (W ⊃ Y) ⊃ Z / Z

1. J ⊃ K 2. ∼J $\lor$ L 3. (K • L) ≡ M -Consider assuming 'J' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. J ⊃ K 2. ∼J $\lor$ L 3. (K • L) ≡ M -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. ∼(N $\lor$ O) 2. P ⊃ ∼Q -Consider assuming 'N $\lor$ P' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. ∼(N $\lor$ O) 2. P ⊃ ∼Q -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. ∼R ≡ S 2. T ⊃ R 3. ∼(R $\lor$ T) ⊃ U -Consider assuming 'S' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. ∼R ≡ S 2. T ⊃ R 3. ∼(R $\lor$ T) ⊃ U -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. A ⊃ (B $\lor$ C) 2. ∼A $\lor$ (D $\lor$ E) 3. ∼(C $\lor$ E) -Consider assuming 'A' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. A ⊃ (B $\lor$ C) 2. ∼A $\lor$ (D $\lor$ E) 3. ∼(C $\lor$ E) -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. F ⊃ [(G $\lor$ H) ⊃ I] -Consider assuming 'F' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. F ⊃ [(G $\lor$ H) ⊃ I] -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

1. J ⊃ (K ⊃ M) 2. L $\lor$ ∼M 3. J $\lor$ (K • ∼L) -Consider assuming 'J' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. J ⊃ (K ⊃ M) 2. L $\lor$ ∼M 3. J $\lor$ (K • ∼L) -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?