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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?