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 determine the truth values of the expressions \( (\sim p \Rightarrow \sim q) \) and \( \sim (\sim p \Rightarrow q) \) when \( p \) and \( q \) are always true, we will follow these steps:

### Step 1: Identify the truth values of \( p \) and \( q \)
Since \( p \) and \( q \) are always true, we have:
- \( p = \text{True} \)
- \( q = \text{True} \)

### Step 2: Calculate \( \sim p \) and \( \sim q \)
Negating \( p \) and \( q \):
- \( \sim p = \text{False} \) (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 truth values:
- \( \sim p = \text{False} \)
- \( \sim q = \text{False} \)

The implication \( \sim p \Rightarrow \sim q \) can be evaluated as follows:
- An implication \( A \Rightarrow B \) is false only when \( A \) is true and \( B \) is false. In all other cases, it is true.
- Here, \( \sim p \) is False and \( \sim q \) is False, so:
- \( \text{False} \Rightarrow \text{False} \) is True.

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

### Step 4: Evaluate the second expression \( \sim (\sim p \Rightarrow q) \)
We already know \( \sim p = \text{False} \) and \( q = \text{True} \).

Now, evaluate \( \sim p \Rightarrow q \):
- \( \sim p \Rightarrow q \) translates to \( \text{False} \Rightarrow \text{True} \).
- This is True (since a False antecedent makes the implication True).

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

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

### Final Results
- The truth value of \( (\sim p \Rightarrow \sim q) \) is **True**.
- The truth value of \( \sim (\sim p \Rightarrow q) \) is **False**.

### Summary of Truth Values
1. \( (\sim p \Rightarrow \sim q) \): **True**
2. \( \sim (\sim p \Rightarrow q) \): **False**