Statements: I – All mangoes are bananas. II – Some bananas are a globe. III –…

Concept:

A syllogism conclusion is valid only when the middle term connecting two statements is distributed (taken in its whole extent) in at least one premise. When the middle term stays undistributed in both premises, no direct relation between the outer terms is guaranteed. If, in that situation, the two candidate conclusions form an exact complementary pair on the same two terms — one saying "some overlap exists" and the other saying "no overlap exists" — then, since these two possibilities exhaust every diagram consistent with the premises, exactly one of them must always be true. The valid reading is then "either conclusion follows".

Application:

1. All mangoes are bananas (mango subset of banana) and All globes are square (globe subset of square) are universal statements; the link between the two chains is the particular premise "Some bananas are a globe" (banana and globe overlap, but not fully).

2. In this chain, the middle term "banana" is the predicate of a universal affirmative in the first statement and the subject of a particular affirmative in the second — undistributed in both. So no definite relation between mangoes and globes, and hence between mangoes and squares, can be fixed from the premises.

3. Conclusion I ("Some mangoes are square") and Conclusion II ("No mango is square") are an exact complementary pair on the same terms (mango, square) — one particular-affirmative, one universal-negative.

4. Because the mango-square relation is undetermined but must be either "some overlap" or "no overlap", one of the two conclusions is always true, even though neither is individually guaranteed by the statements alone.

Cross-check:

Two diagrams both satisfy every statement: placing the mango circle entirely inside the banana-globe overlap makes Conclusion I true; placing it entirely inside the banana region but outside the globe makes Conclusion II true instead. Since both diagrams are equally valid readings of the premises, the certain takeaway is that one of the two conclusions always holds, confirming the either-or reading.

Conclusion:

Either conclusion I or conclusion II follows.