The logical statement `(p to q) vv (q to ~ p)` is :

Options

A. `~p`
B. `~q`
C. t (tautology)
D. F (fallacy)

Explanation

To solve the logical statement `(p to q) vv (q to ~p)`, we will analyze it step by step using a truth table.

### Step 1: Identify the Variables
We have two variables, \( p \) and \( q \).

### Step 2: Create the Truth Table
Since there are two variables, the number of rows in the truth table will be \( 2^2 = 4 \). We will list all possible combinations of truth values for \( p \) and \( q \).

| \( p \) | \( q \) |
|---------|---------|
| T | T |
| T | F |
| F | T |
| F | F |

### Step 3: Calculate \( \neg p \)
We will also need the negation of \( p \).

| \( p \) | \( \neg p \) |
|---------|--------------|
| T | F |
| T | F |
| F | T |
| F | T |

### Step 4: Calculate \( p \to q \)
The implication \( p \to q \) is false only when \( p \) is true and \( q \) is false; otherwise, it is true.

| \( p \) | \( q \) | \( p \to q \) |
|---------|---------|----------------|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |

### Step 5: Calculate \( q \to \neg p \)
The implication \( q \to \neg p \) is false only when \( q \) is true and \( \neg p \) is false; otherwise, it is true.

| \( q \) | \( \neg p \) | \( q \to \neg p \) |
|---------|--------------|---------------------|
| T | F | F |
| F | F | T |
| T | T | T |
| F | T | T |

### Step 6: Combine Results
Now we will combine the results of \( p \to q \) and \( q \to \neg p \) using disjunction (logical OR).

| \( p \) | \( q \) | \( p \to q \) | \( q \to \neg p \) | \( (p \to q) \vee (q \to \neg p) \) |
|---------|---------|----------------|---------------------|-------------------------------------|
| T | T | T | T | T |
| T | F | F | T | T |
| F | T | T | T | T |
| F | F | T | T | T |

### Step 7: Conclusion
Since the final column of the truth table is true for all possible combinations of \( p \) and \( q \), the logical statement `(p to q) vv (q to ~p)` is a tautology.

### Final Answer
The logical statement `(p to q) vv (q to ~p)` is a tautology.

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