https://file.aiquickdraw.com/mathbot/file/1726666924731oafpgab0.jpg

Let's go through each argument step by step to determine which one is not a valid argument.

### Argument A:
1. **Premise 1:** \( \forall x (p(x) \rightarrow q(x)) \)  
   - For every \( x \), if \( p(x) \) is true, then \( q(x) \) is true.
2. **Premise 2:** \( \exists x p(x) \)  
   - There exists some \( x \) such that \( p(x) \) is true.
3. **Conclusion:** \( \therefore \exists x q(x) \)  
   - There exists some \( x \) such that \( q(x) \) is true.

   **Analysis:** This argument is valid. If there exists an \( x \) such that \( p(x) \) is true, and for every \( x \), \( p(x) \) implies \( q(x) \), then there must exist some \( x \) such that \( q(x) \) is true.

### Argument B:
1. **Premise 1:** \( \exists x (p(x) \land q(x)) \)  
   - There exists some \( x \) such that both \( p(x) \) and \( q(x) \) are true.
2. **Conclusion:** \( \therefore \exists x p(x) \)  
   - There exists some \( x \) such that \( p(x) \) is true.

   **Analysis:** This argument is valid. If there exists some \( x \) such that both \( p(x) \) and \( q(x) \) are true, it directly implies that there exists some \( x \) where \( p(x) \) is true.

### Argument C:
1. **Premise 1:** \( \exists x p(x) \)  
   - There exists some \( x \) such that \( p(x) \) is true.
2. **Premise 2:** \( \exists x q(x) \)  
   - There exists some \( x \) such that \( q(x) \) is true.
3. **Conclusion:** \( \therefore \exists x (p(x) \land q(x)) \)  
   - There exists some \( x \) such that both \( p(x) \) and \( q(x) \) are true.

   **Analysis:** This argument is **not valid**. The existence of some \( x \) such that \( p(x) \) is true and the existence of some (potentially different) \( x \) such that \( q(x) \) is true does not guarantee the existence of a single \( x \) where both \( p(x) \) and \( q(x) \) are true.

### Argument D:
1. **Premise 1:** \( \exists x (p(x) \rightarrow q(x)) \)  
   - There exists some \( x \) such that if \( p(x) \) is true, then \( q(x) \) is true.
2. **Premise 2:** \( p(a) \)  
   - \( p(a) \) is true for some specific \( a \).
3. **Conclusion:** \( \therefore q(a) \)  
   - \( q(a) \) is true for the specific \( a \).

   **Analysis:** This argument is valid. If \( p(a) \) is true and there exists some \( x \) (which could be \( a \)) such that \( p(x) \rightarrow q(x) \), then \( q(a) \) must be true.

### Conclusion:
The **invalid** argument is **C**. The premises do not logically lead to the conclusion. 

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

Here are 5 related questions that expand on this topic:
1. What are the rules of inference used to determine the validity of logical arguments?
2. How do existential quantifiers differ from universal quantifiers in logical arguments?
3. Can you explain the difference between validity and soundness in a logical argument?
4. How can we use counterexamples to demonstrate that an argument is not valid?
5. How does the structure of an argument affect its validity?

**Tip:** When dealing with logical arguments, always ensure the conclusion follows directly from the premises by applying the correct rules of inference.

Which of the following arguments is not a valid argument?

Explore which of the four logical arguments, involving universal and existential quantifiers, is invalid. This analysis provides a detailed breakdown of logical inference rules applied in predicate logic, suitable for college-level students.

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