Summary: Because this is complicated. The Big Bad wants PC Abel dead. The Big Bad kidnaps PC Bob's family to coerce Bob into killing Abel. Bob refuses. PC Carlos cuts a different deal with the Big Bad, PC Carlos saying he'll kill Abel if, afterward, the big bad gives Carlos a book about making swords. (Seriously? I have to assume Abel and Carlos? Not good friends.) Also, Carlos's player is terrible at keeping secrets, so the other players know Carlos is gunning for (stabbing for?) Abel.
Talk to the players
The central conflict in my campaigns is the PCs against the universe, not each PC against each other PC. It sounds like the campaign in the question has a different central conflict—and that's totally okay—, but it sounds like this particular conflict isn't the kind of conflict that was supposed to be emphasized.
As the DM, you can go to your toolbox and try to change that conflict by altering the story, introducing other adventuring parties as examples of proper behavior, punishing the players or their PCs for not adhering to your vision, or whatever, but, really, here, I urge trying this tool: talk with the players about your expectations and theirs.
You may want to tell a story of heroic adventurers overcoming great evils. They may want to tell a story of backstabbing ne'er-do-wells that lie, cheat, steal, and murder their way into becoming those great evils. While those two sets of expectations can mesh, they don't often mesh well.
Here's what you can do. Explain that everybody's worked hard to get where they're at for the past eight months and—let's hope, anyway—nobody wants a player-versus-player incident to be the thing everybody remembers about the campaign. Explain that while you'd expected Bob to reject the deal, you hadn't even planned for Carlos to be part of that session, so when Carlos dealt with the Big Bad, you had to improvise, and you screwed up. (Go ahead and shoulder this—the DM's burdens are already vast so this won't add much.) So right now is as good a time as any to discuss expectations—not just with you in real life as the campaign's players but abstractly for their characters in the current campaign as well.
Hammer out those expectations. See what folks want. If everyone had in mind a campaign of backstabbing ne'er-do-wells and the campaign's designed for front-stabbing heroes, adjustments on both sides must be made for the campaign to continue without everyone hating each other. And if a player or two has adopted an adversarial playstyle, urge him to redirect his schemes at NPCs: this may not be as fun for the player as is one-upping the other players, but it means that, when a plan reaches its zenith, the player won't get his car keyed after the screwed player flips the table and leaves. (Note that if the player insists that it's his character—not the player!—who is causing friction, see "My Guy" Syndrome, and if the player always makes nasty characters that can't get along with the party, see here.)
It sounds like the PCs are used to concealing from each other their competing goals. While this is common in media from television to board games, this role-playing game GM's often seen such concealed goals lead to frustration and animosity. That's because, first, the setting is often primarily in the GM's head so that an utterly concealed agenda may fail to reach fruition despite all efforts because the GM doesn't even know the goal exists; second, the only folks who are as familiar with the campaign are the other players, and if the player with a hidden goal can't bounce ideas off them on how to achieve the hidden goal, the player is isolated… in a medium that's all about social interaction.
So have the players and the PCs share their goals at least with the GM, and, quite possibly, with each other. Make it happen in game if you want, discussing hopes and dreams, plans and schemes, over the campfire. I mean, really, the PCs will be camping together until they die—they might as well get used to talking to each other.
Cutting this down: Make the PCs' goals manageable
Below's a brief list of what folks want based on the question.
- Abel: Wants to live.
- Bob: Wants to rescue his family.
- Carlos: Wants to acquire a book about making swords.
Realize that none of these goals mandate Abel's murder by a PC. The only thing mandating Carlos kill Abel is Carlos's semi-secret deal. Take away that secret—and, by the way, that could be a hilarious in-character reveal by PC Carlos—, and then everybody can work together to achieve all three goals.
In fact, were I a player confronted with such disparate goals, I'd recommend faking Abel's death (disguise a corpse and, so he can come along, keep real Abel in, like, a bag of holding wearing a necklace of adaptation) and arranging a meeting with the Big Bad. If the Big Bad keeps his promises—releasing Bob's family and giving Carlos the book—, everybody wins, and adventuring resumes. If the Big Bad breaks his promises, intelligence about the Big Bad's been gained and nothing's lost.
But, either way, the central problem—PC Carlos killing PC Abel—is neatly sidestepped, and the PCs (and their players!) are closer for the experience.