give answer in 2 step with explanation at the end of each step and final answer at the end:Use the eighteen rules of inference to derive the conclusions of the symbolized argument below.Please answer

Question

give answer in 2 step with explanation at the end of each step and final answer at the end:Use the eighteen rules of inference to derive the conclusions of the symbolized argument below.Please answer
in a step by step format showing a rule and reference lines for each step. (example attached)
Use the eighteen rules of inference to derive the conclusions of the symbolized argument below.
B D H
eR EE (EET
MP | MT | HS DS | CD | Simp | Conj | Add | DM | Com | Assoc
| Dist | DN | Trans | Impl | Equv | Exp | Taut | ACP | CP | AP | PP
TO mess
1 ~H>3B
| (pmesase
2 ~H>D
© (remase CONCLUSION
3 ~(B » D) H
FRENISE
4
1.(0 2 R) OS (premise)
2.(P DR) O~ 8 (Premise)
3.] R (Assumption for Indirect Proof - AIP)
4.| RV ~ O (addition, from line 3)
5.1 ~OVR (Commutation, from line 4)
6./ OD R (Material Implication, from line 5)
7.1 S (Modus Ponens, from lines 1and 6)
8. RV ~P (addition, from line 3)
9./ ~PVR (Commutation, from line 8)
10. POR (Material Implication, from line 9)
11. ~ 8 (Modus Ponens, from lines 2 and 10)
12.| Se ~S (Conjunction, from lines 7 and 11)
13. ~ R (Indirect Proof, from lines 3-12)

VIPSolutions · Accepted Answer

Let's proceed step by step, as you requested, for the given argument:

**Premises:**
1. ~H ⊃ B  
2. ~H ⊃ D  
3. ~(B • D)

**Conclusion:**
H

---

### **Step 1: Assume ~H (Indirect Proof / Reductio ad Absurdum)**

- **Assumption for Indirect Proof (AIP):**  
  Assume ~H.

**Explanation:**  
We use indirect proof by assuming the negation of what we want to prove (i.e., assume ~H). If this leads to a contradiction, then H must be true.

---

### **Step 2: Derive B and D Using Modus Ponens, Then a Contradiction**

- **From premise 1 (~H ⊃ B) and our assumption (~H):**  
  By Modus Ponens, we get B.
- **From premise 2 (~H ⊃ D) and our assumption (~H):**  
  By Modus Ponens, we get D.
- **Now, combine B and D:**  
  By Conjunction, B • D.
- **But premise 3 says ~(B • D):**  
  We have a contradiction.

**Explanation:**  
Assuming ~H leads us to both B and D, which gives us (B • D). But premise 3 says ~(B • D) is true, so our assumption (~H) must be false.

---

### **Final Answer**

#### **Therefore, H is true.**

**Summary:**  
- **Step 1:** Assume ~H (AIP).
- **Step 2:** Derive B and D using premises 1 and 2, leading to a contradiction with premise 3.
- **Conclusion:** H (by Indirect Proof).

---

**Reference Lines:**
1. ~H ⊃ B (Premise)
2. ~H ⊃ D (Premise)
3. ~(B • D) (Premise)
4. ~H (Assumption for Indirect Proof)
5. B (Modus Ponens, 1, 4)
6. D (Modus Ponens, 2, 4)
7. B • D (Conjunction, 5, 6)
8. ~(B • D) (Premise, 3)
9. Contradiction (7, 8)
10. H (Indirect Proof, 4–9)

---

**Conclusion:**  
**H** is the valid conclusion.

VIPSolutions · Answer

# Deriving the Conclusion Using Rules of Inference

## **Premises**
1. $ \sim H \supset B $  
2. $ \sim H \supset D $  
3. $ \sim (B \cdot D) $

## **Conclusion**
H

---

### **Step 1: Assume $ \sim H $ (Indirect Proof / Reductio ad Absurdum)**

- **Line 4:** Assume $ \sim H $.

**Explanation:**  
We begin by assuming the negation of what we wish to prove (i.e., $ \sim H $). If this assumption leads to a contradiction, we can conclude that H must be true.

---

### **Step 2: Derive B and D, Leading to a Contradiction**

- **Line 5:** From premise 1 and assumption $ \sim H $:  
  By **Modus Ponens**, we conclude $ B $ (from lines 1 and 4).
  
- **Line 6:** From premise 2 and assumption $ \sim H $:  
  By **Modus Ponens**, we conclude $ D $ (from lines 2 and 4).

- **Line 7:** Combine B and D:  
  $ B \cdot D $ (by **Conjunction**, from lines 5 and 6).

- **Line 8:** Reference premise 3:  
  $ \sim (B \cdot D) $ (premise).

- **Line 9:** Contradiction:  
  We have $ B \cdot D $ and $ \sim (B \cdot D) $. This is a contradiction.

**Explanation:**  
The assumption $ \sim H $ allows us to derive both $ B $ and $ D $, leading to $ B \cdot D $. However, since premise 3 states $ \sim (B \cdot D) $, this creates a contradiction, indicating that our initial assumption must be false.

---

### **Final Answer**

#### **Therefore, H is true.**

**Summary of Steps:**
- **Step 1:** Assume $ \sim H $ (AIP).
- **Step 2:** Derive $ B $ and $ D $ using premises 1 and 2, leading to a contradiction with premise 3.

---

### **Reference Lines**
1. $ \sim H \supset B $ (Premise)
2. $ \sim H \supset D $ (Premise)
3. $ \sim (B \cdot D) $ (Premise)
4. $ \sim H $ (Assumption for Indirect Proof)
5. $ B $ (Modus Ponens, 1, 4)
6. $ D $ (Modus Ponens, 2, 4)
7. $ B \cdot D $ (Conjunction, 5, 6)
8. $ \sim (B \cdot D) $ (Premise, 3)
9. Contradiction (7, 8)
10. H (Indirect Proof, 4–9)

---

**Conclusion:**  
**H** is the valid conclusion.

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