[Jimbee (OP)]

“Your actions are highly unethical!”

—Mr. Spock, The Empath, Star Trek: The Original Series,
 Season 3, Episode 12, Original air date December 6, 1968.

As the above quote shows, the fictional Mr. Spock at least was very moral.

But could a purely logical person actually be so? How would he probably behave? Or could we even know?

This is psychological question. But I suppose it could be a larger philosophical one too.

[alancalverd]

As the above quote shows, the fictional Mr. Spock at least was very moral.

No, it shows that he assumes the other person to understand the word "ethical".

I can point out the unchristian behavior of a christian without subscribing to his superstitions, just as I could find someone guilty of blasphemy without having any interest in the subject at all.

[Bored chemist]

In general society tries to arrange that "crime doesn't pay".
If they more or less succeed in that, anyone who is logical will follow the ethics/ morals of that society in order to avoid consequences of not doing so.
Anyone strictly logical will typically act in a moral manner for that reason.

[Petrochemicals]

Logically I think everyone over 45 should be terminated A La Logans Run. Morally there is an argunent for it too, taking up recourses etc.