Statements: All purple are orange. All black are orange. Conclusions: I. Some…

Concept

When two premises share the same predicate as 'All A are C' and 'All B are C', the statements only place A and C, and B and C, inside a common circle — they never state any direct relationship between A and B. If two proposed conclusions about A and B form a complementary pair (one says 'Some B are A', the other says 'No B is A'), then in every diagram consistent with the premises exactly one of the pair must be true, so the valid inference is 'Either … or …' follows.

Application

Here, 'All purple are orange' and 'All black are orange' place both purple and black inside the orange circle, with no statement linking purple and black to each other directly. Conclusion I ('Some black are purple') and Conclusion II ('No black is purple') are exactly such a complementary pair — one claims some overlap between the black and purple regions, the other claims none.

1. Draw purple and orange (All purple are orange) and black and orange (All black are orange) as two circles both inside the orange circle.

2. Case A — let the purple and black circles overlap inside orange: this diagram satisfies both premises, and in it 'Some black are purple' is true.

3. Case B — let the purple and black circles sit inside orange without touching: this diagram also satisfies both premises, and in it 'No black is purple' is true.

4. No premise rules out either case, so neither conclusion holds by itself in every diagram — but between Case A and Case B, one of the two conclusions is always true.

Cross-check

Testing every diagram consistent with the two statements (overlapping circles, non-overlapping circles, or one circle nested fully inside the other) always lands in either the 'some overlap' picture or the 'no overlap' picture — never both false and never a case outside these two. That confirms the complementary-pair rule and rules out 'only I', 'only II', and 'neither' as the option to choose.