91Ó°ÊÓ

First, create a table with all possible combinations of truth values for p, q, and r. There are 3 variables, so we will have 2^3 = 8 rows in our table.

Now, we will fill in the table columns with the given premises and conclusion. 1. The first premise is simply \(p\). 2. The second premise is \(p \rightarrow q\). This is true when either \(p\) is false or \(q\) is true. 3. The third premise is \(\sim q \vee r\). This is true when either \(q\) is false or \(r\) is true. 4. The conclusion is \(r\). The completed truth table looks like this: | p | q | r | p | p→q | ∼q∨r | r | |:-:|:-:|:-:|---|-----|------|---| | T | T | T | T | T | T | T | | T | T | F | T | T | F | F | | T | F | T | T | F | T | T | | T | F | F | T | F | T | F | | F | T | T | F | T | T | T | | F | T | F | F | T | F | F | | F | F | T | F | T | T | T | | F | F | F | F | T | T | F |

In the above truth table, the columns that represent the premises are the columns for p, p→q, and ∼q∨r. The column that represents the conclusion is the column for r.

An argument is valid if, whenever all the premises are true, the conclusion is also true. In our truth table, we can see there are two rows where all premises are true: 1. Row 1: p, p→q, and ∼q∨r are true and the conclusion r is also true. 2. Row 5: p, p→q, and ∼q∨r are false, but the conclusion r is true. Since in both cases, the conclusion (r) is true when all the premises are true, this argument form is valid.