Achievments for Victory?

Your exact example I can't think of any instances of, but plenty of similarities with other mission-card games. For instance, Mission: Red Planet. It uses mission cards as one way to get victory points, but you only reveal them at the end like in Ticket to Ride. Neither of these is revealed as you go, and neither of these has a "get three out of 4 to win."

I think if you are making the win condition by 3/4 cards, I'm more inclined to let other players see them so they can balance blocking other players vs. achieving their own victory. If they are hidden, it'll depend on how clear the objectives are on each mission card. If other players would have no way of really seeing what you were building for, then keeping them hidden means all the players will be inclined to act independently - and not build counter strategies.

In your case, with 20 some mission cards, I'm betting they'll either be:
- variations on a few themes (in which case they could be hidden, because the other players would know which theme you are playing towards, even if they don't know the particulars), or
- 20 wildly different missions, in which case I think they should be shown

As sure as i am of that, I'm also sure there is a good case for an opposite approach. :)

If players don't need to accomplish all the tasks on the card, and they don't need to complete all cards for victory, it sounds like you could simplify things but just figuring out how many tasks you want them to accomplish, and making one card per task.

For instance, you list three possible tasks, so let's assume it's "achieved" if you hit two of them, and you mention achieving three cards out of four for a win, so you're talking about completing six tasks out of a possible twelve. Why not just hand out twelve cards, each with a single task? It will be much easier for players to keep track of which things they've achieved and which they haven't.

Otherwise there's going to be a lot of game time spent checking which sub-task has been achieved; additionally, players can memorize 20 cards with three tasks each which will defeat the purpose of keeping them secret (which I like).

In your 'defeat an army' example, there's no real point in hiding the cards as the other players will likely figure out that's your goal when they see you claim a token for having wiped out an enemy army ^_^.

You seem to be pretty smitten with the idea of using the chips, but if I could offer another possible solution, this time to the defeated army scenario. In Memoir 44, defeated units are kept track of by keeping aside one figure from the defeated unit as a 'trophy' worth VP. You could, similarly, have players keep one figure from a defeated army every time, weather or not plat player has defeating armies as a goal. that way, you can prevent players from cheating by slyly moving chips off one card to another (^_^) while still being able to prove the player actually has defeated X armies.

Otherwise, I would reveal all objective cards so other players can keep track of how many chips everyone has on each objective in order to stave off cheating.

A game that some friends and i play quite often is a smaller wargame called Nexus Ops.
In that game you have a hand of cards that are victory conditions, each giving an amount of points.
The first to 10 points is the winner, and its fun to watch certain moves be more agressive than others in effort to gain these points :)