Question 1171515: Hey, I hope you are doing well. I have these homework questions that I can't figure out.
A, ¬F → ¬A ⊢ D → (¬E → F)
I'm supposed to construct a formal proof
Thank you!
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Alright, let's break down this logical entailment proof. We need to show that from the premise A, ¬F → ¬A, we can derive the conclusion D → (¬E → F).
Here's a step-by-step proof using natural deduction:
**1. A (Premise)**
**2. ¬F → ¬A (Premise)**
**3. ¬¬A (Double Negation Introduction, 1)**
**4. A (Double Negation Elimination, 3)**
**5. F (Modus Ponens, 2, 4)**
**6. ¬E → F (Conditional Introduction, 5)** - Here, we discharge the assumption of ¬E.
**7. D → (¬E → F) (Conditional Introduction, 6)** - Here, we discharge the assumption of D.
**Explanation:**
1. We start with the given premises, A and ¬F → ¬A.
2. From A, we can introduce a double negation (¬¬A).
3. We can eliminate the double negation to get A.
4. Now, we have A, and ¬F → ¬A. Since A is true, ¬A is false. Therefore, ¬F must be false, which means F is true.
5. We've derived F.
6. To prove ¬E → F, we assume ¬E. Since we've already derived F, the implication ¬E → F is true. We discharge the assumption ¬E.
7. To prove D → (¬E → F), we assume D. Since we've already derived ¬E → F, the implication D → (¬E → F) is true. We discharge the assumption D.
**Therefore, A, ¬F → ¬A ⊢ D → (¬E → F) is a valid entailment.**
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