Let's break down the problem you uploaded.

Statements:

1. If she likes cheeseburgers, she will not go out to eat.

   • Let $p$ represent the statement "she likes cheeseburgers."
   • Let $q$ represent the statement "she will go out to eat."

   The conditional statement translates to: $p \rightarrow \neg q$ (If $p$, then not $q$).

2. She will go out to eat.

   • This is $q$, meaning: $q$

3. She hates cheeseburgers.

   • This means "she does not like cheeseburgers," or: $\neg p$

Translation of the argument into symbolic form:

1. The argument starts with the conditional: $p \rightarrow \neg q$ (If she likes cheeseburgers, she will not go out to eat).

2. The second statement is: $q$ (She will go out to eat).

3. The conclusion is: $\neg p$ (She does not like cheeseburgers).

Validity:

The argument seems to follow a pattern similar to Modus Tollens, which is of the form:

• $p \rightarrow \neg q$
• $q$
• Therefore, $\neg p$.

This is a valid form of argument.

Answer:

• The argument is valid.
• In symbolic form, the argument is: $(p \rightarrow \neg q), q \vdash \neg p$

Would you like more details or clarifications on the reasoning?

Here are five questions to help expand understanding:

1. What is Modus Tollens, and why is it a valid form of argument?
2. How can we prove an argument is invalid using a truth table?
3. What other valid forms of arguments are there in logic?
4. How does the negation of a statement affect the logical structure of an argument?
5. Can a valid argument have false premises and still be valid?

Tip: Always translate each part of a problem into symbolic form first—it helps simplify complex arguments into manageable logic steps!