The argument presented involves three components:

1. $p \vee q$ (p OR q)
2. $q$ (q is true)
3. Conclusion: $\therefore \sim p$ (Therefore, p is false)

We will construct a truth table to test the validity of this argument.

Step 1: List all possible truth values for $p$ and $q$.

Step 2: Analyze the argument.

• Premise 1: $p \vee q$ is true except in the last row.
• Premise 2: $q$ is true in rows 1 and 3.
• The conclusion $\sim p$ (p is false) is true in rows 3 and 4.

Step 3: Validate the argument.

For the argument to be valid, whenever both premises ($p \vee q$ and $q$) are true, the conclusion $\sim p$ must also be true.

• In row 3, both premises are true ($p \vee q = T$ and $q = T$), and the conclusion $\sim p$ is also true.
• In row 1, the premises are true, but the conclusion $\sim p$ is false.

Since there is a case where the premises are true, but the conclusion is false, the argument is invalid.

Would you like further details or have any questions about this process?

Here are 5 related questions for practice:

1. How do you construct a truth table for a more complex statement involving conjunctions and implications?
2. Can an argument be valid if the conclusion is false in some cases?
3. What happens if we switch $p$ and $q$ in the argument?
4. How can a truth table be used to check logical equivalence?
5. What does it mean for an argument to be "logically valid"?

Tip: In logical arguments, always focus on rows where all premises are true to determine validity.