what are the truth values of ` ( ~ p Rightarrow ~ q) and ~( ~ p Rightarrow q)` respectively, when p and q always speak true in any argument ?

Options

A. T, T
B. F, F
C. T, F
D. F, T

Answer

To solve the problem, we need to evaluate the truth values of the expressions \( ( \sim p \Rightarrow \sim q) \) and \( \sim ( \sim p \Rightarrow q) \) given that both \( p \) and \( q \) are always true.

### Step 1: Determine the truth values of \( p \) and \( q \)
Since it is given that \( p \) and \( q \) always speak true, we have:
- \( p = \text{True} \)
- \( q = \text{True} \)

### Step 2: Evaluate \( \sim p \) and \( \sim q \)
The negations of \( p \) and \( q \) are:
- \( \sim p = \text{False} \) (since \( p \) is True)
- \( \sim q = \text{False} \) (since \( q \) is True)

### Step 3: Evaluate the first expression \( ( \sim p \Rightarrow \sim q) \)
Using the implication truth table:
- The implication \( A \Rightarrow B \) is false only when \( A \) is true and \( B \) is false; otherwise, it is true.

Now substituting \( \sim p \) and \( \sim q \):
- \( \sim p \Rightarrow \sim q \) becomes \( \text{False} \Rightarrow \text{False} \).

According to the truth table:
- \( \text{False} \Rightarrow \text{False} = \text{True} \)

So, the truth value of the first expression \( ( \sim p \Rightarrow \sim q) \) is **True**.

### Step 4: Evaluate the second expression \( \sim ( \sim p \Rightarrow q) \)
First, we need to evaluate \( \sim p \Rightarrow q \):
- Substituting the values, we have \( \text{False} \Rightarrow \text{True} \).

According to the truth table:
- \( \text{False} \Rightarrow \text{True} = \text{True} \)

Now we need to negate this result:
- \( \sim (\text{True}) = \text{False} \)

So, the truth value of the second expression \( \sim ( \sim p \Rightarrow q) \) is **False**.

### Conclusion
The truth values of the expressions are:
- \( ( \sim p \Rightarrow \sim q) = \text{True} \)
- \( \sim ( \sim p \Rightarrow q) = \text{False} \)

Thus, the final answer is:
- True and False.