To construct a formal proof using a statement/reason table for the argument:

Given premises:

1. $g \rightarrow \sim n$
2. $(p \vee q) \rightarrow r$
3. $r \rightarrow w$
4. $g \vee p$

Conclusion: $n \rightarrow w$

Let's construct the proof in a structured format.

Statement/Reason Table

Step	Statement	Reason
1	$g \rightarrow \sim n$	Premise
2	$(p \vee q) \rightarrow r$	Premise
3	$r \rightarrow w$	Premise
4	$g \vee p$	Premise
5	Assume $n$	Assumption for Conditional Proof
6	$\sim g$	Modus Tollens on Step 1 and 5
7	$p$	Disjunctive Syllogism on Step 4 and 6
8	$p \vee q$	Addition on Step 7
9	$r$	Modus Ponens on Step 2 and 8
10	$w$	Modus Ponens on Step 3 and 9
11	$n \rightarrow w$	Conditional Proof from Steps 5–10

Explanation

In this proof, we assume $n$ to derive $w$ using the premises given. Here's how it unfolds:

• Steps 1–4 list the premises.
• Step 5 assumes $n$ for the purpose of a conditional proof.
• Step 6 uses $g \rightarrow \sim n$ and our assumption of $n$ to conclude $\sim g$ (if $g$ were true, $n$ would not be true).
• Step 7 applies disjunctive syllogism to the statement $g \vee p$, concluding $p$.
• Step 8 uses addition to state $p \vee q$, which lets us apply the implication $(p \vee q) \rightarrow r$ to get $r$ in Step 9.
• Steps 10 and 11 conclude with $w$ from $r \rightarrow w$ and complete the proof that $n \rightarrow w$.

This formal proof shows that $n \rightarrow w$ is logically derived from the given premises.

Would you like further details or have any questions?

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1. How can we use conditional proofs to derive implications?
2. What role does Modus Ponens play in deriving conclusions?
3. How does Disjunctive Syllogism simplify complex arguments?
4. What is the purpose of assuming a statement in a proof?
5. Why is Addition important in formal logic proofs?

Tip: Conditional proofs often start with assuming the antecedent to derive the consequent, helping structure complex arguments logically.