Test the validity of the argument:
p → q
q → r
Therefore, ~ r → ~p

Test the validity of the argument:
p → q
q → r
Therefore, ~ r → ~p

Invalid

Valid

To test the validity of the argument, we need to check whether the conclusion \( \neg r \rightarrow \neg p \) necessarily follows from the premises \( p \rightarrow q \) and \( q \rightarrow r \). This is known as the **contrapositive** form of the implications.

### Argument:
1. \( p \rightarrow q \)
2. \( q \rightarrow r \)
3. Therefore, \( \neg r \rightarrow \neg p \)

We can analyze this using logical reasoning or truth tables.

### Step 1: Analyze the premises

1. **Premise 1: \( p \rightarrow q \)**  
   This means that if \( p \) is true, then \( q \) must also be true.

2. **Premise 2: \( q \rightarrow r \)**  
   This means that if \( q \) is true, then \( r \) must also be true.

### Step 2: Derive the conclusion

We are asked to derive the conclusion \( \neg r \rightarrow \neg p \). This is the contrapositive form of the original implications.

- The contrapositive of \( p \rightarrow q \) is \( \neg q \rightarrow \neg p \).
- The contrapositive of \( q \rightarrow r \) is \( \neg r \rightarrow \neg q \).

Now, let's connect these:

1. From the contrapositive of \( q \rightarrow r \), we have \( \neg r \rightarrow \neg q \).
2. From the contrapositive of \( p \rightarrow q \), we have \( \neg q \rightarrow \neg p \).

By combining these two contrapositive forms, we can derive:

- \( \neg r \rightarrow \neg q \) (from \( q \rightarrow r \))
- \( \neg q \rightarrow \neg p \) (from \( p \rightarrow q \))

Thus, \( \neg r \rightarrow \neg q \rightarrow \neg p \) implies \( \neg r \rightarrow \neg p \), which is exactly the conclusion.

### Conclusion: The argument is **valid**.

Would you like a further breakdown of this or have any questions?

Here are 5 related questions to explore further:
1. What is the contrapositive of an implication, and how does it relate to the original statement?
2. Can you provide other examples where contraposition is used to validate arguments?
3. How would a truth table confirm the validity of this argument?
4. What is the difference between a contrapositive and a converse statement in logic?
5. How can understanding contrapositives help in proving mathematical theorems?

**Tip:** Remember, the contrapositive of a logical implication is always logically equivalent to the original statement.

Explore the validity of logical implications with the problem: p → q, q → r. Check if the conclusion ~r → ~p logically follows from the given premises. Understand the use of contrapositive in logical reasoning and truth tables. Suitable for advanced high school and above.

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