Syllogism • Topic 4 of 4

Either–Or Cases

When two conclusions individually do not follow but together cover all cases (same subject–predicate, one positive and one negative), the answer is 'either I or II follows'.

When the answer is "Either I or II"

Sometimes neither conclusion follows on its own, yet together they cover every possibility. That is the signature of an either–or case. Two conditions must both be met:

  • The two conclusions share the same subject and predicate (e.g., both are about "A and B").
  • They are complementary — one positive, one negative — so at least one must always be true (e.g., "Some A are B" and "No A is B").
Quick test. If you find yourself saying "in some diagrams conclusion I is true and in the others conclusion II is true, but never both fail together" — that is exactly an either–or.

The classic complementary pairs

Conclusion IConclusion II
Some A are BNo A is B
All A are BSome A are not B

Note: if both conclusions follow definitely, it is not either–or (the answer would be "both follow"). Either–or applies only when neither follows alone.

✅ Solved examples

1. Conclusions: I. Some A are B. II. No A is B. Neither alone follows. What is the answer?
They are complementary and exhaust all cases → Either I or II follows.
2. I. All A are B. II. Some A are not B. (complementary pair) — answer?
One must hold in any diagram → Either I or II.
3. Why is "Either" used here?
Because the pair covers every possibility, so at least one is always true even though neither is individually certain.
4. I. Some A are C. II. Some A are not C. Answer if neither alone follows?
Complementary and exhaustive → Either I or II.

✏️ Practice — try these, take hints as needed

1. I. Some X are Y. II. No X is Y. Neither alone follows — answer?
Complementary pair.
—
—
Either I or II follows
2. Are "All A are B" and "Some A are not B" complementary?
One negates the other.
—
—
Yes
3. Either-or needs the conclusions to share what?
Same subject & predicate.
—
—
Same subject–predicate
4. If both conclusions follow definitely, is it "either-or"?
Either-or needs neither alone.
—
—
No (both follow)
5. I. Some A are B. II. Some A are not B. Answer if neither alone follows?
Exhaustive complementary pair.
—
—
Either I or II follows

📝 Topic test — 8 questions

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