Test the validity of the argument:
p → (q → ~r)
(q ∧ ~r) ∨ r
~q
Therefore, p

Invalid

Valid

\

To test the validity of the argument, we can use truth tables or logical reasoning. Here's the argument:

1. \( p \rightarrow (q \rightarrow \neg r) \)
2. \( (q \land \neg r) \lor r \)
3. \( \neg q \)
4. Therefore, \( p \)

We'll check if the conclusion \( p \) necessarily follows from the premises. 

### Step 1: Analyze the premises

1. \( p \rightarrow (q \rightarrow \neg r) \) can be written as:
   - If \( p \) is true, then \( q \rightarrow \neg r \) must be true.

2. \( (q \land \neg r) \lor r \) can be true if either \( q \) and \( \neg r \) are both true, or \( r \) is true.

3. \( \neg q \): \( q \) is false.

### Step 2: Derive the conclusion

Given \( \neg q \) (from the third premise), we know that \( q \) is false. 

Now, let's consider the second premise \( (q \land \neg r) \lor r \):
- Since \( q \) is false, the left part \( q \land \neg r \) becomes false, so for the premise to be true, \( r \) must be true.

Thus, \( r \) is true.

Now, consider the first premise \( p \rightarrow (q \rightarrow \neg r) \):
- Since \( q \) is false, \( q \rightarrow \neg r \) is true regardless of \( r \).
- Therefore, \( p \rightarrow \text{True} \), meaning the implication is always true regardless of the value of \( p \).

Since the premises are true but do not logically force \( p \) to be true, the conclusion \( p \) is not necessarily true.

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

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to explore further:
1. How do truth tables help in determining the validity of arguments?
2. What are the rules for implications in logic?
3. Can you create a truth table for this argument to confirm the conclusion?
4. How does negation affect the truth values in logical statements?
5. What are some common logical fallacies that lead to invalid arguments?

**Tip:** When analyzing logical arguments, carefully evaluate each premise and how they connect to the conclusion.

Explore the validity of logical arguments using truth tables and logical reasoning. Analyze the argument p → (q → ~r), (q ∧ ~r) ∨ r, ~q, Therefore, p. Discover why the conclusion p is invalid based on the given premises.

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