`(p hArr q) hArr (p implies q) ^^` ........

Answer

To solve the expression `(p ↔ q) ↔ (p → q)`, we need to analyze the logical equivalences involved.

### Step-by-Step Solution:

1. **Understand the Symbols**:
- `p ↔ q` (p is equivalent to q): This means that both p and q have the same truth value. It is true if both are true or both are false.
- `p → q` (p implies q): This means that if p is true, then q must also be true. It is false only when p is true and q is false.

2. **Break Down the Expression**:
- The expression `(p ↔ q) ↔ (p → q)` states that the equivalence of p and q is equivalent to the implication of p by q.

3. **Evaluate `p ↔ q`**:
- The truth table for `p ↔ q` is as follows:
- p = True, q = True → `p ↔ q` = True
- p = True, q = False → `p ↔ q` = False
- p = False, q = True → `p ↔ q` = False
- p = False, q = False → `p ↔ q` = True

4. **Evaluate `p → q`**:
- The truth table for `p → q` is:
- p = True, q = True → `p → q` = True
- p = True, q = False → `p → q` = False
- p = False, q = True → `p → q` = True
- p = False, q = False → `p → q` = True

5. **Combine the Results**:
- Now we need to evaluate `(p ↔ q) ↔ (p → q)` based on the results from the previous steps.
- We will create a truth table for the entire expression:
- For each combination of p and q, we will check the values of `p ↔ q` and `p → q`, and then determine the value of `(p ↔ q) ↔ (p → q)`.

6. **Truth Table for the Entire Expression**:
| p | q | p ↔ q | p → q | (p ↔ q) ↔ (p → q) |
|-------|-------|-------|-------|---------------------|
| True | True | True | True | True |
| True | False | False | False | True |
| False | True | False | True | False |
| False | False | True | True | True |

7. **Final Result**:
- The expression `(p ↔ q) ↔ (p → q)` evaluates to True for the combinations (True, True), (True, False), and (False, False), and False for (False, True).

### Conclusion:
The final expression `(p ↔ q) ↔ (p → q)` is not a tautology, as it does not evaluate to True for all possible truth values of p and q.