Statements : No women teacher can play. Some women teachers are athletes.…

Concept

In categorical syllogisms, a conclusion 'follows' only when it is a NECESSARY logical consequence of the statements — not merely something that could also happen to be true. Two rules decide necessity: (1) if either premise is negative, any valid conclusion must also be negative — a premise that only denies an ability can never necessitate a positive claim of that ability; and (2) a valid conclusion can never introduce a term absent from both premises. A conclusion that is merely POSSIBLE (not contradicted by the premises) but not FORCED by them does not follow.

Application

1. Statement 1 ('No women teacher can play') is a universal negative linking women teachers with the ability to play; Statement 2 ('Some women teachers are athletes') is a particular affirmative linking women teachers with athletes.

2. Conclusion I talks about 'male athletes' — a term present in neither statement — so it fails the no-new-term rule outright and cannot be deduced.

3. Converting Statement 2 gives 'Some athletes are women teachers'. Combining this with the negative Statement 1 forces only a NEGATIVE conclusion about athletes — 'some athletes are not able to play' — this is the one thing the statements actually guarantee.

4. Conclusion II claims the POSITIVE 'some athletes can play'. The statements never establish this — they say nothing about any athlete outside the women-teacher group — so it is at best a possibility, never a necessary consequence, and hence does not follow.

Cross-check

Picture three circles: 'women teachers' and 'able to play' never overlap (Statement 1), while 'women teachers' and 'athletes' partly overlap (Statement 2). Every athlete inside that overlap is, by Statement 1, forced outside the 'able to play' circle — so the only guaranteed fact is that some athletes cannot play. Whether any athlete OUTSIDE the women-teacher circle can play is left completely open by this picture — a possible case, not a certainty — so Conclusion II is not forced true. Conclusion I's 'male athletes' has no circle at all here, so it cannot even be checked.

• 'Only conclusion I follows' fails because 'male athletes' never appears in the statements.

• 'Only conclusion II follows' fails because the statements never force the positive claim 'some athletes can play' — they force only its negative counterpart.

• 'Either I or II follows' applies when two conclusions share the same subject and predicate as an affirmative/negative pair; conclusion I ('male athletes') and conclusion II ('athletes') do not share a common subject, so the either/or format itself does not fit this pair.

• 'Neither I nor II follows' matches the derivation: conclusion I fails the no-new-term rule, and conclusion II is only a possible, not a necessary, consequence of the statements.

Result

Since Conclusion I introduces an undefined term and Conclusion II is only a possible — not a necessary — consequence of the statements, neither conclusion validly follows: the correct choice is 'Neither I nor II follows'.