Given below are two premises (A and B). Four conclusions are drawn from them.…

Concept — A proposition of the form "Most X are Y" says that more than half of X lies inside Y. It fixes no proportion whatever for the group Y, since Y may be very much larger than X, so under literal majority semantics it converts only to "Some Y are X" and never to "Most Y are X". This examination, however, sets such statements in the traditional treatment: each "most" premise is read as a particular affirmative that converts simply, so a premise taken individually licenses exchanging its own subject and predicate, while two particular premises taken jointly license nothing at all, because the middle term is nowhere distributed and the two majorities need not overlap.

Application — apply that treatment to each of the four conclusions in turn.

1. Premise (A), "Most of the dancers are physically fit", is the only premise containing both the term dancer and the term physically fit. Exchanging its subject and predicate gives "Most of the physically fit persons are dancers", which is conclusion (c), drawn from premise (A) taken individually.

2. Premise (B), "Most of the singers are dancers", is the only premise containing both the term singer and the term dancer. Exchanging its subject and predicate gives "Most of the dancers are singers", which is conclusion (b), drawn from premise (B) taken individually.

3. Conclusion (a), "Most of the singers are physically fit", pairs singer with physically fit. No single premise contains both of those terms, so it can be reached only by joining premise (B) to premise (A) through the middle term dancers.

4. Conclusion (d), "Most of the physically fit persons are singers", pairs physically fit with singer, again one term from each premise, so it needs that same joint chain and then a further exchange of subject and predicate.

5. Both premises are particular, so the joint chain is barred, and only the two single-premise conversions stand: (b) and (c).

Cross-check — counting shows both why the joint chain fails and why the treatment above has to be stated openly. Take 100 dancers, 60 of them physically fit, and 100 singers, 60 of whom are dancers. Those 60 singer-dancers may be the 40 dancers who are not physically fit together with 20 who are, leaving only 20 of the 100 singers physically fit: both premises still hold while (a), and with it (d), fails. The same counting also shows that a reversed majority is not a theorem. If there were 1,000 physically fit persons in all, the 60 physically fit dancers would be a mere 6 per cent of them, so (c) does not follow under literal quantifier semantics either.

Validly drawn conclusions under the examination convention and its official key: (b) and (c). Read simple conversion of a "most" statement as the convention this question is set in, not as a general law of logic.