Let's complete the proof step by step and include the missing details. The goal is to show the argument's conclusion step by step using logical rules.

We are given:

1. $(p \lor r) \rightarrow q$ (Hypothesis)
2. $q \rightarrow t$ (Hypothesis)
3. We need to prove $t$.

Proof:

Explanation:

• In step 2, we used Addition to infer $p \lor r$ from $p$. This is a valid step because $p \lor r$ is true if $p$ is true.
• In step 4, we used Modus Ponens on steps 2 and 3. Since we know $p \lor r$ is true (from step 2) and we have $(p \lor r) \rightarrow q$, we can conclude that $q$ is true.
• In step 6, we again used Modus Ponens on steps 4 and 5. Since $q$ is true and $q \rightarrow t$, we conclude that $t$ is true, which is the conclusion of the argument.

Would you like further details on any step?

Follow-up questions:

1. Can you explain why Modus Ponens is used in this proof?
2. What is the purpose of the Addition rule in step 2?
3. Why is $p \lor r$ valid from the hypothesis of $p$?
4. How does the structure of this proof ensure the conclusion $t$?
5. What would happen if we reversed the hypotheses of $q$ and $t$?

Tip:

When working with logical proofs, always identify which statements are your hypotheses and carefully apply rules like Modus Ponens or Addition based on earlier steps.