(B) is incorrect because we cannot conclude that most Bs are Cs (B-most-C). All we can conclude from the premises is that some Bs are Cs (B-some-C). Here's an example of why:
Let's say we have 100 not-As. if most not-As are Cs, then at least 51 of them (and up to 100) are Cs (first premise).
Relating this to the contrapositive of second premise (not A - >B) then all 100 not-As are Bs. But what if the total number of Bs is 10,000? It could be, or it could be 100, or it could be another number. Because we do not know the number of Bs, we cannot say with certainty that most Bs are Cs (B-most-C, which is answer (B)). All we can say is that some Bs are Cs (B-some-C, which is answer (A)).
LamontJanuary 19, 2022
I am a bit confused when it comes to quantifiers. This question premise says not A most of not C the contrapositive is not C some not A. I initially wrote the contrapositive as some however, decided to change it back to most. Thank goodness my answer was correct as the answer choice used some. This brings to the same issue with the conclusion and its contrapositive. I noticed this argument flowed using the contrapositive of not C some Some A and the contrapositive of the second premise concluded C some B. Is it correct to the contrapositive of the premise some most to some or vice versa? Then the conclusion contrapositive remains the same as the given conclusion. I hope this is is not confusing.
AbigailJanuary 24, 2022
Hello @lamont,
I think you might have read the premise wrong. I see (Not A-most-C), not (Not A-most-Not C). The reversal of the premise is (C-some-Not A). I always recommend that my students not think of the "reversals" of quantifying statements as "contrapositives" because they follow completely different rules as the contrapositives of conditional statements. It is best to separate mentally the two forms of reversals. You were initially right to reverse the first most statement to "some," as all "most" statements need to be reversed to "some" statements. Here's an example why: Let's say we have 50 apples, 30 of which are not red apples. We can say (Apples-most-Not Red) and we can reverse that to (Not Red-Some-Apples). Because we don't know how many "not red" things there are, we cannot say that most "not red" things are "apples." We could have 100 "not red" things, thereby the 30 not red apples constitutes less than half (some) of the totality of "not red" things. When reversing quantifying statements (All, Most and Some), they all need to be reversed down to "some statements."
The correct answer choice was deduced by taking the contrapositive of the second statement (it is a conditional statement, so it has a contrapositive) (Not A -->B) with the reversal of the first premise (C-some-Not A). Thereby concluding that (C-some-B) or its reversal (B-some-C).
I hope this clarifies your question. Feel free to follow-up if you are still unsure.
Abigail
LamontJanuary 25, 2022
Thank you, Abigail. You actually have helped me understand quantifiers greatly.
RaviFebruary 6, 2022
Happy you're understanding them really well now!
RaviFebruary 6, 2022
Happy you're understanding them really well now!
MelissaEngelkingMarch 8, 2024
Why would the contrapositive of the first premise be C-some-A verses not C-some-A?
Emil-KunkinApril 8, 2024
I think the correct interpretation here would be C some Not A. We want to keep the terms the same here, so we are still talking about C and Not A. We know that the vent diagram of C and Not A includes some overlap, so it must be the case that some Cs are also not A.
