It may be easier to verbalize than we think. I just realized that for me the trick might be to not be too abstract and formal when first thinking it out, but look at it more like an everyday problem.

Instead of this:

"Given two appointments, one that starts at time A and ends at time B and another that starts at time C and ends at time D, how can I tell if they conflict?"

More like this:

"I need to run another errand while I'm in town for the interview. I can either do it after the interview, or if I do it first, I'd better be done before the interview."

That gives me something concrete to think about so I can make sure I am on the right track and won't miss my interview! And from there it's a pretty easy path to working out the detailed formula. If I jump into the abstract description right away I'm a lot more likely to confuse myself.