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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 \( q \).

| \( p \) | \( q \) | \( p \vee q \) | \( \sim p \) |
|:-------:|:-------:|:--------------:|:------------:|
|   T     |   T     |       T        |      F       |
|   T     |   F     |       T        |      F       |
|   F     |   T     |       T        |      T       |
|   F     |   F     |       F        |      T       |

### 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.

Use a truth table to test the validity of the argument: p ∨ q, q, ∴ ~p.

Learn how to test the validity of a logical argument using a truth table with propositions p ∨ q, q, and the conclusion ∴ ~p. This problem explores key concepts in propositional logic and truth tables, suitable for students in Grades 9-12.

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