https://file.aiquickdraw.com/mathbot/file/1726865372874orjlw07c.jpg

Let's break down the problem based on the image.

The problem is asking you to translate a logical argument into symbolic form and then determine whether it is valid or invalid.

### The argument is:

1. **If she wears perfume, she will not find a spouse.**
2. **She will find a spouse.**
3. **Therefore, she does not wear perfume.**

### Symbolic Translation:
Let \( p \) represent "she wears perfume" and \( q \) represent "she will find a spouse."

- The first statement, "If she wears perfume, she will not find a spouse," translates to:
  \[
  p \to \neg q
  \]
- The second statement, "She will find a spouse," translates to:
  \[
  q
  \]
- The conclusion, "She does not wear perfume," translates to:
  \[
  \neg p
  \]

So the argument structure is:

\[
\left( p \to \neg q \right), \, q \, \therefore \, \neg p
\]

### Validity:
This is an instance of **Modus Tollens**:
- Modus Tollens is the rule of inference stating: If \( p \to \neg q \), and \( q \), then \( \neg p \).
- This form is logically valid.

### Answer:
- The argument is **valid**.
- The symbolic form of the argument is \( \left( p \to \neg q \right), \, q \, \therefore \, \neg p \).

Would you like further details or have any questions?

Here are some related questions:
1. How does Modus Tollens differ from Modus Ponens?
2. Can you give an example of an invalid argument form?
3. What are other common valid argument forms in propositional logic?
4. How does the truth table for \( p \to \neg q \) look?
5. What is the role of negation in logical arguments?

**Tip:** Understanding common valid argument forms like Modus Ponens and Modus Tollens helps in quickly recognizing the structure of logical reasoning.

Translate each argument into symbolic form. Determine whether the argument is valid or invalid. If she wears perfume, she will not find a spouse. She will find a spouse. Therefore, she does not wear perfume.

This problem involves translating a logical argument into symbolic form and determining its validity using the Modus Tollens inference rule. Explore how to analyze logical statements, including conditionals and negations, in a structured symbolic form.

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