I'm not sure I understand the original premise correctly, but if it's "most A's are B's" and "most B's are C's", then we can't say absolutely anything about the relationship between A and C. You can have, say, 5 of the A's. 3 of those (most) are B's. But what if there are a total of 100 B's, of which 51 (most) are C's, there's no requirement for those 51 to include the 3 A's. Example: most young women in China (A) are bike riders (B). Most bike riders (B) in China are men (C). These can both be correct, as we know there are more men than young women in China. Yet the conclusion "some young women in China are men" (some A are C) is clearly not correct.
Yes, if most A are B AND most A are C, then some B are C would be correct. That's different from the original question, at least the way I understood it.
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You can have, say, 5 of the A's. 3 of those (most) are B's. But what if there are a total of 100 B's, of which 51 (most) are C's, there's no requirement for those 51 to include the 3 A's.
Example: most young women in China (A) are bike riders (B). Most bike riders (B) in China are men (C). These can both be correct, as we know there are more men than young women in China. Yet the conclusion "some young women in China are men" (some A are C) is clearly not correct.
Apologies if I misread the original premise.