Question 1164909: i have been having a hard time answering this rules of inference and rules of replacement, can you help me answer this?
1) ﬢ[P→ﬢ(Q۸R)]
ﬢ[P→ﬢ
________________
∴P۸(Q۸R)
2) 1. (P۷Q)→(R۸S)
2. P
_________________
∴ R
3) 1. (P۷Q) ۷ ﬢR
2. ﬢP ۸ ﬢQ
__________________
∴ﬢR ۷ S
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! Rules of inference and replacement can feel like a complex puzzle at first, but once you identify the "shape" of the statements, the rules become much clearer. Here are the step-by-step proofs for your three problems.
---
### 1) Prove:
**Given:**
| Step | Statement | Rule Used |
| --- | --- | --- |
| 1 | | Premise |
| 2 | | **Material Implication (Impl)** on 1 |
| 3 | | **De Morgan's (DM)** on 2 |
| 4 | | **Double Negation (DN)** on 3 |
**Explanation:** To break into a negated conditional, we first turn the "if-then" into an "or" statement, then use De Morgan's to distribute the negation.
---
### 2) Prove:
**Given:** 1.
2.
| Step | Statement | Rule Used |
| --- | --- | --- |
| 1 | | Premise |
| 2 | | Premise |
| 3 | | **Addition (Add)** on 2 |
| 4 | | **Modus Ponens (MP)** on 1, 3 |
| 5 | | **Simplification (Simp)** on 4 |
**Explanation:** Since we have , we can "add" anything to it using an "or" statement. This satisfies the left side of the conditional in Step 1, allowing us to extract the right side.
---
### 3) Prove:
**Given:**
1.
2.
| Step | Statement | Rule Used |
| --- | --- | --- |
| 1 | | Premise |
| 2 | | Premise |
| 3 | | **De Morgan's (DM)** on 2 |
| 4 | | **Disjunctive Syllogism (DS)** on 1, 3 |
| 5 | | **Addition (Add)** on 4 |
**Explanation:** In Step 3, we used De Morgan's to show that neither nor is true. This contradicts the first part of Premise 1, which forces to be true. Once we have , we can use Addition to add any variable (in this case, ).
---
### Logic Tool Reference
A helpful tip for these is to always look at your **Conclusion** first. If it has a "" (like in problem 3), you often only need to find one side of it and use **Addition**.
Would you like to try another set, or should we go deeper into how **De Morgan's Law** works?
|
|
|
