That's... a good question. I'm sure someone will test it before the game comes out, but I have doubt that it actually will work without Enemy Within, considering that it seems to be one of the achievements being added BECAUSE of Enemy Within.
Due to technical difficulties, the light at the end of the tunnel has been shut off...
Yeah, there are a couple new achievements listed that sound like things we could get, now, but I'd be shocked if they'd work without Enemy Within, since they were added with all of the others that are clearly tied to the upcoming expansion.
Given the nature of the changes being made to the game, they seem like the sort of things people would at least like the option to opt out of (I could definitely see people getting upset if they permanently lose vanilla XCOM), so I think the real question is whether or not the achievements will require you to be playing with the Enemy Within content or if they'll work if you simply have it installed.
(Yes, this is the exact same post that I just deleted, but I wanted to remove some "new" observations that were already made by someone else over a day ago in another topic I somehow completely missed, and I'm past the time period when GameFAQs would let me edit the post.)
I'm noticing that research time is longer on classic mode, so some of the changes seems to have taken effect already. Fighting with 4 people past month 5 sure is... different than usual. Going to post here if the achievement works.