Why would you allow it?
Unless the game is bent on the players killing one another, as in an all evil party where everyone is trying to become the most powerful and take over the area: like a mob kind of game.
I did that, I ran an evil game where all the players had to work together, and, at the same time, look for a way to take each other out. Everyone knew it, and it was fun that way...I was the medium, I knew all the behind the scenes actions (there was alot of secret notes) and the players had to put outside knowledge away, (but they were good players so it wasn't hard) and at the end of the game, while it was fun, most didn't want to play it again. It was more or less..." lets play an all evil party" kind of thing, once over, it was done.
But, in a game, player's shouldn't be allowed to kill one another, that just opens to many doors to bad blood, and out of game drama. As the DM I think you should step in. Its a game, you can stop the game, or end it completly if you have to so that the players GROW UP and decided that it shouldn't be done. Also, if killing that player is the only option, then your a bad DM...if that player is a bad gamer, messes with peoplel, or other such things, then he should be made to leave. ANd once gone, the story moves on without him.
Also, yes, a player may get it in mind that he could slip a knife into another player, but then I'd think most players would be like "what the hell are you doing? why would u want to do that?" I think the other players would step up and convince him that what he was doing was stupid and childish.
The DM is there to keep these "out of place reactions" out of the game.
Just cause a player could do it, doesn't mean he would, and just cause he could come up with a valid reason, doesn't mean it would really happen. I think you have to look at the real reason why there PK's, most arn't for story reasons, or because "thats what my charcter would do" reasons.
Game ON