I have problem with the following syllogism.
Some A are B All B are C ———————————— Some B are C Some A are C
I think both are correct answers while my book says that the only correct answer is "some A are C".
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It depends on what kind of logic you are studying. In Aristotelean term logic, "all B are C" has existential import, which is to say it is only true if there are B's. Hence it follows by immediate inference that "some B are C". If the point of the exercise is to ask what can you infer from both premises that does not simply follow by immediate inference from either one, then "some A are C" does this but not "some B are C". We would need to see exactly how the question is worded in order to judge whether that is the best answer.
In predicate logic, "all B are C" does not have existential import, which is to say it is trivially true in the event that there are no B's. Hence "some B are C" does not follow from it. To deduce "some B are C" you would need to reason from "some A are B" to "there exists at least one B" and then from this together with "all B are C" to "there exists at least one thing that is B and C".
So for myself, preferring the predicate logic approach, I would say you are correct that both "some B are C" and "some A are C" follow.