Statements: All bags are cakes. All lamps are cakes. Conclusions: Some lamps…

Concept: A direct categorical conclusion linking two extreme terms is valid only when their shared "middle" term is distributed (refers to ALL of its members) in at least one premise. When both premises are "All X are Y" (A-type) with the SAME term as their predicate, that predicate stays undistributed in both, so no single definite conclusion about the two extreme terms can be drawn.

Separately, when two offered conclusions about the same pair of extreme terms are exact contradictories — a particular affirmative ("Some X are Y") against the flat universal negative on the same terms ("No X is Y") — logic guarantees at least one of the pair is true, even though neither is individually provable on its own. This is the "either/or" case.

Application:

1. Premise 1, "All bags are cakes", is an A-type statement: it distributes "bags" (the subject) but leaves "cakes" (the predicate) undistributed.

2. Premise 2, "All lamps are cakes", is also A-type: it distributes "lamps" but again leaves "cakes" undistributed.

3. "Cakes" is the shared middle term here, and it is undistributed in BOTH premises — so no direct, certain relationship between "bags" and "lamps" can be validly derived from the premises alone.

4. Conclusion I, "Some lamps are bags", cannot be certified true on its own: proving a definite overlap needs the shared term distributed somewhere, which it is not.

5. Conclusion II, "No lamp is bag", cannot be certified true on its own either, for the identical reason — a flat denial of any overlap is just as unsupported by an undistributed middle.

6. Compare I and II as a pair rather than individually: "Some lamps are bags" (particular affirmative) and "No lamp is bag" (universal negative) share the exact same two terms and are exact contradictories of one another — they cannot both be false at the same time.

7. Since a genuine contradictory pair always has at least one true member, and no third option is offered, the either/or rule settles it: "Either I or II follows".

Cross-check: Build two concrete groupings consistent with both premises. Grouping A — bags = {b1, b2}, lamps = {l1, l2}, all of them inside cakes, with bags and lamps kept disjoint: here conclusion II ("No lamp is bag") holds and I is false. Grouping B — bags = {b1, b2}, lamps = {b1, l2}, sharing member b1, all inside cakes: here conclusion I ("Some lamps are bags") holds and II is false. Both groupings satisfy the two given premises, and in each one exactly one of I/II turns out true — confirming the either/or verdict rather than a fixed single conclusion or a blanket “neither”.

Result: Either I or II follows.