The Hostage Fallacy is not a real logical fallacy, but it comes close:

You have a hostage at gunpoint. I have you at gunpoint. We assume both of us can fire one shot. You can shoot the hostage. I can shoot you. If I shoot you while you have your gun pointed at the hostage, you will be able to shoot the hostage too. If you switch to point your gun at me, I can shoot you before you can shoot anybody. I will not kill you if you release the hostage. Your priority is to live.

You have four choices:

  1. Shoot the hostage.
  2. Don't shoot the hostage (but stay where you are).
  3. Switch your gun to point at me.
  4. Let the hostage go.

3. will lead to you being shot, and the hostage and me will be unharmed. You WILL die.

4. leaves you alive, the hostage alive, and me alive.

1. and 2. seem the obvious choices. However:

If you choose 1., you now have no hostage. I will kill you. You and the hostage will die. You WILL die.

If you choose 2., you now have no hostage. If you do not shoot the hostage, I can shoot you, and because you are not shooting the hostage, only you die. You WILL die.

QED, if you want to live, you choose 4 and let the hostage go.

Ergo, the hostage fallacy is that taking a hostage did you no good at all.