>I see. One day you sat down and wrote up the truth table for p -> q >all by yourself. Then you added in a few more connectives, and set >about proving soundness and completeness. Good job! We'll just clone >your logic gene and achieve propagation that way.
Truth table validity for p -> q is NOT intuitive nor does anybody bother with it in normal mathematical proof; you assume p and confirm q, let alone informal usage.
By contrast, you have to be carefully taught to give a T to "If all dogs are white, then my dog is black" because the antecedent is false.