The following is an argument from Patrick Suppes' Introduction to Logic:
My first issue is that I don't understand why the author included columns $2$ and $5$. Taking every truth value in the table as correct, aren't columns $1, 3, 4, 6$ sufficient for showing this is not a truth-functional connective?
My second issue is with the truth values in the table itself. If we know that $M$ is false, then to me it makes no sense to claim that $\diamondsuit M$ is true. This would make sense if $M$ was "It is possible that there COULD be life on Mars". But $M$ is just "There is life on Mars."
$\newcommand{\poss}{\Diamond} \newcommand{\necs}{\Box} \newcommand{\sem}[1]{\mbox{ $[\![ #1 ]\!]$}}$
(I'll be using $N$ instead of $W$ throughout my post in order to avoid confusion with the set of possible worlds $W$ defined below.)
As far as I see it, it would indeed have sufficed to compare only the truth values of $M/\poss M$ and $N/\poss N$ respecitively, as there exists no function which maps $F$ to $T$ (in the case of $M$) but at the same time $F$ to $F$ (in the case of $N$).
Presumably the reason why $\neg M/\poss \neg M$ was included was for the conclusion that
thereby demonstrating an attempt to indeed analyze $\poss$ as truth-functional, to then show how $N/\poss N$ makes this analysis fail, which makes the impossibility of such an analysis more evident than just saying "There can not be any such function".
So I assume the two additional colums were included for didactical reasons, but with respect to what is sufficient to show the non-truth-functionality of $\poss$, contrasting the values for $\poss M$ and $\poss N$ would have sufficed.
Qiaochu Yuan already brought it well to the point, but in order to make it better understandable, I'll explain it a bit more formally:
In order to define the meaning of the modal operators $\necs$ ("necessary") and $\poss$ ("possible") in possible world semantics, a model $\mathcal{M} = \langle W, R, V \rangle$ for modal propositional logic will consist of
Propositions will now be evaluatied not only relative to an assignemt function, but with respect to a possible world in which this proposition is supposed to be true or false.
The idea is that $\poss P$ ("It is possible that $P$) is true if, in the world which we evaluate a proposition in, we can imagine an alternate world in which that proposition could be true, even though it might not be true in our actual world.
Correspondingly, $\necs P$ ("It is necessary that $P$) will be true if $P$ is true in every possible world we can reach, or conversely, we can not imagine a world in which $P$ is not true.
The concept of imaginability is formally caputed by the accessibility relation $R$. For the evaluation of modal propositions, we need to find proofs in worlds that our current world can reach:
If a proposition $P$ is true in none but one world which, however, is a world that we can not imagine (i.e. the accessibility relation does not include the pair of our world and the alternative one), we think it not possible that $P$ is true.
Likewise, it might happen that there are worlds in which $P$ is not true but in our world we still deem $P$ a necessity because the worlds in which $P$ does not hold are out of reach for our mental accessibility.
The formal definitions for the modal operators are then quite straightforward:
W.r.t. the Mars example, in order for $\poss M$ to be true in our world (call it $w$), it suffices to show that we can access some other world $w'$ in which $M$ is true. Intuitively, it should be easily possible to imagine a world in which there is life on Mars, this world being mentally acessible from ours - regardless of whether the same proposition is a true fact at our own world. But then, we found a $w' \in W$, $wRw'$ such that $M$ holds at $w'$, and hence, $\poss M$ evaluates as true at our actual world $w$.
At the same time, $\poss \neg M$ is true as well: We can equally well imagine a world in which there is no life on Mars (most prominently ours, if one sets up $R$ such that $w$ stands in an accessibility realtion to itself), so there is a world $w'$, $wRw'$ in which $\neg M$ holds, and therfore both $\poss M$ and $\poss \neg M$ are true in the world $w$ under consideration.
This is not the case for $N$. One would assume that a world in which a mathematically absurdity like $2+2=5$ is a true statement is not a world we can ever access, so there is no world $w' \in W$ such that $wRw'$ and $N$ is true at $w'$, and therfore $\poss N$ is false at $w$.
Correspondingly, $\necs \neg N$ would be a true proposition in our world $w$, as in any world we can reach from $w$, the negation of $2+2=5$ holds.
This explains why the author assumes that both $\poss M$ and $\poss \neg M$ are true propositions in our world, while $\poss N$ is not.