Let's walk through completing the truth table for the statement $(\neg p \land (p \rightarrow q)) \rightarrow \neg q$.

Steps to break down the expression:

1. Negation of $p$: $\neg p$
2. Implication $p \rightarrow q$: This is false only when $p = T$ and $q = F$, and true otherwise.
3. Conjunction: $\neg p \land (p \rightarrow q)$ combines the results from steps 1 and 2.
4. Negation of $q$: $\neg q$
5. Final Implication: The overall expression $(\neg p \land (p \rightarrow q)) \rightarrow \neg q$ is evaluated by combining the conjunction from step 3 with the negation from step 4.

Let's fill in the missing values step by step:

1. Row 1 ($p = T, q = T$):

• $\neg p = F$
• $p \rightarrow q = T$
• $\neg p \land (p \rightarrow q) = F \land T = F$
• $\neg q = F$
• $(\neg p \land (p \rightarrow q)) \rightarrow \neg q = F \rightarrow F = T$

2. Row 2 ($p = T, q = F$):

• $\neg p = F$
• $p \rightarrow q = F$
• $\neg p \land (p \rightarrow q) = F \land F = F$
• $\neg q = T$
• $(\neg p \land (p \rightarrow q)) \rightarrow \neg q = F \rightarrow T = T$

3. Row 3 ($p = F, q = T$):

• $\neg p = T$
• $p \rightarrow q = T$
• $\neg p \land (p \rightarrow q) = T \land T = T$
• $\neg q = F$
• $(\neg p \land (p \rightarrow q)) \rightarrow \neg q = T \rightarrow F = F$

4. Row 4 ($p = F, q = F$):

• $\neg p = T$
• $p \rightarrow q = T$
• $\neg p \land (p \rightarrow q) = T \land T = T$
• $\neg q = T$
• $(\neg p \land (p \rightarrow q)) \rightarrow \neg q = T \rightarrow T = T$

Filled Truth Table:

Now the table is complete.

Would you like further details or explanations on any part of this truth table?

5 Follow-up Questions:

1. What is the truth value of an implication when the premise is false?
2. How does negation affect the values of a logical statement?
3. Why does the conjunction in row 3 result in a false implication?
4. What would change if the expression was $(p \land (p \rightarrow q)) \rightarrow q$?
5. How would you construct a truth table for a biconditional statement $p \leftrightarrow q$?

Tip: The implication $p \rightarrow q$ is true unless $p$ is true and $q$ is false.