> "Do or you can't" means that if you Do you can

You have this backwards IMO. In logical terms (not vernacular usage), "Do or you can't" being true means that if you CAN (play), then you do/did the chores. It definitely does not preclude other don'ts leading to you can'ts.

It only makes "do" a necessary (not sufficient) condition for "can".

Another way to see it : "Do or you can't" is a form of "A or B", so "you can" means B is false, so A must be true.

You sure? I am pretty sure you need something like, "Child! If you do not finish your chores, I will not allow you to use the PC to do anything other than read the HowTo.com webpages on how to do housechores. Ergo, you will be unable to play Minecraft."