I've been gluing and internally patching tires for pretty much ever. If the cut got enough of the cords, you'll know soon enough-- the cut will start to spread and you won't be able to stop it. Most nicks and small slices can be "repaired" with just superglue from the outside, and sometimes by gluing a patch to the inside of the tire.

The only thing that will almost certainly trash a tire is a cut in the sidewall. I've seen stitch jobs, but that's too far even for me.