Summary:
Logical implication cannot be defined by merely a truth table. The idea logical implication can be defined through a truth table is a misconception many logicians have been fooled into believing.
Details:
P → Q does not mean ~(P ^ ~Q).
This is because if ~(P ^ ~Q), then it is not necessarily the case that P → Q. For example, if it is not the case that “Tom eats a slice of pizza and he did not buy the slice of pizza,” it does not follow that “If Tom eats a slice of pizza, then he bought the slice of pizza.” Tom could have stole the slice of pizza or made it himself. The statement “It is not the case that ‘Tom eats a slice of pizza and he did not buy the slice of pizza’” is a statement about a single case (~(P ^ ~Q), where P and Q have a defined truth value), while the statement “If Tom eats a slice of pizza, then he bought the slice of pizza” is a general statement about all cases (P → Q, where P and Q DO NOT have defined truth values). We cannot infer from a single case what is true about all cases.
P → Q actually means “In every case that P is true, Q is also true.” This is logically equivalent to “There is no case in which P is true and Q is false,” and logically equivalent to “In every case, ~(P ^ ~Q) is true.”
A proposition P may not be true in all cases. So obviously, the compound proposition ~(P ^ ~Q) may not be true in all cases. This comes to show that ~(P ^ ~Q) is a statement about a single case. It is not a general statement about all cases. So from the single case ~(P ^ ~Q), we cannot prove the general claim P → Q.
In conclusion, P → Q and ~(P ^ ~Q) are NOT logically equivalent. P → Q DOES NOT mean ~(P ^ ~Q).