David, the second case you mention [BAC, ????, therefore BID] is in fact a valid syllogism if you add, as the second premise, CAD.
Rule 6 on distribution goes from bottom (conclusion) to top (premises), not the other way around. What rule 6 prohibits is having a term distributed in the conclusion and not distributed in the premise in which it occurs. There’s no problem having a term distributed in a premise and not distributed in the conclusion. Roughly, the idea is this.
If in my premises I haven’t been using a term to refer to all the things that it can refer to, it’s hard to see how I can start doing so in my conclusion. On the other hand, it’s perfectly in order to use a term to refer to all of the things it can refer to in a premise, and then not to do so in the conclusion. You can validly go from all to some, but not from some to all.
So, if you check the complete syllogism according to the six rules, you get:
Rule 1 satisfied (at least one universal)
Rule 2 satisfied (vacuously – no particular premise so no need to check further)
Rule 3 satisfied (at least one affirmative premise)
Rule 4 satisfied (vacuously – no negative premise so no need to check further)
Rule 5 is satisfied (the middle term, C, is distributed in the second premise)
Rule 6 is satisfied (vacuously, since no term in the conclusion is distributed)
I hope this clears things up?
If you have any other queries or questions, please do let me know. And if you start a thread, please send your initial post directly to me, otherwise I might not see it for some time.