I need to solve a proof and I cannot figure it out. The instructions say only that I will have to use subproofs within subproofs.
Premises:
A or B
A or C
Conclusion:
A or (B and C)
To solve this proof using subproofs, we will need to consider different cases and use subproofs to derive the desired conclusion. Here's an explanation of the steps you can follow to solve this proof:
1. Start by assuming "A" as a subproof to consider one of the cases.
2. With the assumption of "A," you have what you need to conclude the desired statement: "A or (B and C)." So you can use the disjunction introduction (∨I) rule to derive the conclusion within this subproof.
3. Now, assume "B" as another subproof to consider the second case.
4. With the assumption of "B," you can go further and assume "C" as yet another subproof.
5. Using these assumptions, you can derive (B and C) using the conjunction introduction (∧I) rule.
6. With (B and C) derived, you can use the disjunction introduction (∨I) rule to conclude "A or (B and C)" within the subproof.
7. Next, you will need to derive the desired conclusion using the assumption "C" from the subproof where you assumed "B."
8. Using the assumption "C," you can once again conclude (B and C) using the conjunction introduction (∧I) rule.
9. Finally, you can use the disjunction introduction (∨I) rule to conclude "A or (B and C)" within the subproof where you assumed "B."
10. After deriving "A or (B and C)" within both subproofs, you can then use the disjunction elimination (∨E) rule to combine the two subproofs and derive the desired conclusion on a broader level.
Remember to properly introduce and discharge assumptions, as well as apply the correct rules of inference along the way. This method of using subproofs within subproofs will help you systematically derive the conclusion "A or (B and C)" from the given premises "A or B" and "A or C."