To solve this, we need to analyze the given expression $p \to (p \to q)$ and determine when it is false.

1. First, recall the truth table for the implication $p \to q$:
   • $p \to q$ is false only when $p$ is true and $q$ is false.
   • Otherwise, $p \to q$ is true.

Now, we want $p \to (p \to q)$ to be false. This means:

• $p$ must be true (since $p \to (p \to q)$ is false only when the antecedent is true and the consequent is false).
• $(p \to q)$ must be false for the entire implication to be false.

For $(p \to q)$ to be false:

• $p$ must be true and $q$ must be false.

Thus, the truth values are:

• $p = \text{true}$
• $q = \text{false}$

Conclusion:

For $p \to (p \to q)$ to be false, $p$ must be true and $q$ must be false.

Would you like to see the detailed truth table or explanation for the implication?

Here are 5 related questions:

1. What is the truth table for $p \to q$?
2. How does double implication (biconditional) differ from implication?
3. Can an implication be false when the consequent is true?
4. How would you express $p \to (p \to q)$ using logical operators like AND and OR?
5. In which logical systems does implication behave differently?

Tip: When analyzing implications, focus on when the antecedent is true and the consequent is false—this is the only time the implication is false!