The meaning of Implication in Logic

If you start out with a false premise, then, as far as implication is concerned, you are free to conclude anything. (This corresponds to the fact that, when $P$ is false, the implication $P \rightarrow Q$ is true no matter what $Q$ is.)

If you start out with a true premise, then the implication should be true only when the conclusion is also true. (This corresponds to the fact that, when $P$ is true, the truth of the implication is the same as the truth of $Q$.)

Not really as an answer but as an anecdote I'll sketch the following real life situation (in abstract terms to avoid controversy).

A politician $s$ declares, in a menacing voice: "if we would do $P$ then $Q$ will ensue!" where $P$ is something like electing his adversary, or not adopting the Draconic measures he proposes, and $Q$ are catastrophic events like people losing their jobs and the country plunging into a deep crisis. Now suppose $s$ is lucky and manages to avoid $P$, but that then $Q$ happens anyway. Now does this show that $s$ lied? Since $P$ is false and $Q$ is true, we are in the second line of your table, you can read off that $P\Rightarrow Q$ is deemed true in this case. In fact since $s$ prophesized about a circumstance $P$ that did not happen, later events could not have shown him a liar either way. An this in spite of the fact that by common sense the statement he made was either false (if doing $P$ would actually have prevented $Q$) or irrelevant (if $Q$ would have happened independently of $P$, or depending on other conditions than those of $P$).

You see how smart politicians are? (Of course $s$ can be shown to be a liar if he does not manage to avoid $P$, but then being out of office anyway, $s$ probably won't care much about being proven a liar as well.)

• If $P$ then $Q$ is equivalent to $P\to Q$
• $P$ only if $Q$ is also equivalent to $P\to Q$ (See example here)
• $Q$ if $P$ is same as if $P$ then $Q$ and equivalent to $P\to Q$

However, note (following statement which is not given in the original question): $P$ if $Q$ is equivalent to if $Q$ then $P$ or $Q\to P$