I have problem with the following syllogism.
Some A are B
All B are C
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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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1Of course, Some B are C follows from All B are C... But the syllogism derive a conclusion from both premsies. The rules for syllogism states that the "middle" term (B) will not be present in the conclusion, that must "specify" the relation betwee the "extremes" (A and C).– Mauro ALLEGRANZANov 25, 2016 at 11:15
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So what is the correct answer? I am confused by what the middle term in general is– Jerry KNov 25, 2016 at 11:18
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The correct answer is "Some A are C". See Syllogism : Examples : Darii.– Mauro ALLEGRANZANov 25, 2016 at 11:23
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some b are c is correct, but not as the conclusion of a syllogism, which involves 3 propositions.– user20153Nov 25, 2016 at 22:51
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Jerry K-- The middle term is the term common to both premises. It connects the two, and enables a conclusion larger than each premise individually.– Mark AndrewsNov 28, 2016 at 20:25
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3 Answers
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.
I would reach that problem this way:
Some A are B
All B are C (so, some A are the same as B, this means some A are C )
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Some B are C - cannot be the right answer as it is clearly said previously that all B are C
What's the question to which the book says there's only one correct answer? Given two premises, there are typically a number of different (mutually compatible) conclusions that can be validly derived from them. "Some B are C" follows from "All B are C"--so that makes the first premise irrelevant. But an argument with redundant premises can still be valid. But if you're looking at a series of problems to be solved in a logic textbook, there must be some instructions preceding the series of problems, and what exactly are they asking you to do with these sets of premises? (i.e. What's the question to which the book says there's only one correct answer?)
