Answer the following questions
Examine whether the statement pattern `[p rarr (~q vv r)] harr ~[p rarr (q rarr r)]` is a tautology, contradiction or contingency.

Answer

To determine whether the statement pattern \([p \rightarrow (\sim q \vee r)] \leftrightarrow \sim [p \rightarrow (q \rightarrow r)]\) is a tautology, contradiction, or contingency, we will construct a truth table and analyze the results step by step.

### Step 1: Identify the Components
We have three variables: \(p\), \(q\), and \(r\). We will also need the negation of \(q\) (\(\sim q\)), and the implications \(p \rightarrow (\sim q \vee r)\) and \(p \rightarrow (q \rightarrow r)\).

### Step 2: Create the Truth Table
We will create a truth table with all possible combinations of truth values for \(p\), \(q\), and \(r\). There are \(2^3 = 8\) combinations.

| \(p\) | \(q\) | \(r\) | \(\sim q\) | \(\sim q \vee r\) | \(p \rightarrow (\sim q \vee r)\) | \(q \rightarrow r\) | \(p \rightarrow (q \rightarrow r)\) | \(\sim [p \rightarrow (q \rightarrow r)]\) | \([p \rightarrow (\sim q \vee r)] \leftrightarrow \sim [p \rightarrow (q \rightarrow r)]\) |
|-------|-------|-------|------------|---------------------|------------------------------------|---------------------|-------------------------------------|--------------------------------------------|---------------------------------------------------------------------------------------------------|
| T | T | T | F | T | T | T | T | F | F |
| T | T | F | F | F | F | F | F | T | F |
| T | F | T | T | T | T | T | T | F | F |
| T | F | F | T | T | T | F | F | T | T |
| F | T | T | F | T | T | T | T | F | F |
| F | T | F | F | F | T | F | T | F | T |
| F | F | T | T | T | T | T | T | F | F |
| F | F | F | T | T | T | F | T | F | T |

### Step 3: Analyze the Final Column
Now we will look at the last column of the truth table, which represents the expression \([p \rightarrow (\sim q \vee r)] \leftrightarrow \sim [p \rightarrow (q \rightarrow r)]\).

The values in the last column are:
- F
- F
- F
- T
- F
- T
- F
- T

### Step 4: Conclusion
Since the last column contains both true (T) and false (F) values, the statement is neither a tautology (which would require all true values) nor a contradiction (which would require all false values). Therefore, it is classified as a **contingency**.