This image shows a logical proof problem. The proof is asking for the missing statement in line 3, and you are provided with several options to choose from. Let's break it down step by step:

Given Proof:

1. [q]
   This is the assumption where $q$ is introduced.

2. $(p \lor q) \rightarrow (p \land (q \rightarrow r))$
   This is a logical implication involving $p$, $q$, and $r$. It might have been derived from earlier logical steps or premises.

3. ???
   We need to find the missing statement.

4. $p \land (q \rightarrow r)$
   This is derived in some way. It seems to be a conjunction, possibly obtained from the implication in line 2.

5. $q \rightarrow r$
   A conditional that is related to the conjunction in line 4.

6. $r$
   Here $r$ is obtained, likely from a combination of previous statements.

7. $((p \lor q) \rightarrow (p \land (q \rightarrow r))) \rightarrow r$
   This is a more complex implication involving the earlier steps. This looks like it's concluding a larger argument.

8. $q \rightarrow (((p \lor q) \rightarrow (p \land (q \rightarrow r))) \rightarrow r)$
   This is the final step, concluding that $q$ leads to the result in line 7.

Options:

• $p \lor q$: Disjunction of $p$ and $q$.
• $p \land q$: Conjunction of $p$ and $q$.
• $p$: Just $p$ alone.
• $q \rightarrow r$: Conditional $q$ implies $r$.

Logical Structure:

The goal is to fill in the missing line with something that logically follows from the earlier steps.

In line 2, we have an implication: $(p \lor q) \rightarrow (p \land (q \rightarrow r))$. This could suggest that in line 3, we are moving toward asserting either $p \land (q \rightarrow r)$, or some step that leads us to that.

Looking at line 4, we directly have $p \land (q \rightarrow r)$. This implies that the missing step (line 3) should lead us toward that conjunction. The options provided suggest that we should introduce the left side of the implication from line 2, which is $p \lor q$, allowing us to apply modus ponens.

Answer:

Thus, the missing step is $p \lor q$, allowing the implication from line 2 to lead us to line 4 through modus ponens.

Would you like a deeper explanation of how each line connects? Here are some questions to further explore the topic:

1. Why is $p \lor q$ the most suitable step in line 3?
2. How does modus ponens apply between lines 2 and 4?
3. What role does the assumption $q$ play in the overall proof?
4. How do you handle proofs involving nested implications?
5. How does the final step relate to the earlier lines in terms of logical structure?

Tip: When working with logical proofs, it's crucial to identify patterns like modus ponens and how they help you derive conclusions step by step.