11.1 Syllogisms, Sets & Possibility Conclusions

Translate statements into sets

A syllogism asks what follows from supplied premises. Treat the category names as abstract labels, even when they are familiar railway words. An imagined fact about the outside world cannot become an extra premise. A conclusion follows definitely only when it is true in every arrangement of sets satisfying the premises.

“Some” means at least one; it does not mean only some. A statement that some A are B is compatible with all A being B. Universal statements do not by themselves prove that either category contains a member. If a question expressly adopts a different existence convention, follow its stated convention, but do not introduce an unstated witness in ordinary set reasoning.

“Only” reverses the subset direction

Only A are B means all B are A. For example, “Only approved items are dispatched items” places dispatched items inside approved items. It does not say all approved items are dispatched, and it does not forbid dispatched items from belonging to a third category such as fragile items.

Draw B inside A. If another premise puts some A inside C, B might overlap that part or might not. The subset statement alone cannot answer whether any B are C. In particular, do not invent an “exclusive boundary” keeping B out of every other set. That would be a new premise.

In the conventional aptitude wording used in the exercises here, only a few A are B means both “some A are B” and “some A are not B.” It therefore rules out all A being B. It does not rule out all B being A. Keep these two statements separate on your diagram so the negative part is not lost.

Definite conclusions and possible conclusions

A definite conclusion must survive every valid diagram. To disprove it, produce one valid counterexample. A possibility conclusion requires at least one valid diagram supporting it. A possible overlap is not a definite overlap; a definite overlap also satisfies logical possibility.

Consider: Some A are B. All B are C. No C is D. The known A–B member must be C and cannot be D. Therefore, some A are not D follows definitely. The remaining members of A are unconstrained: some might be D or all might remain outside D. Neither “all A are C” nor “no A is D” follows just from the premises.

If a possibility is proposed, construct it explicitly. “Some A being D is possible” can be satisfied by adding a separate A member in D while keeping the known A–B–C member outside D. “All B being D is possible” fails because the existing B member is inside C and C is disjoint from D.

Worked example: the negative witness matters

Premises: Only a few Locomotives are Electric. All Electric are High-Speed. No High-Speed is Steam.

Conclusion I says some Locomotives are not Steam. The positive part of “only a few” supplies a Locomotive–Electric member. That member is High-Speed and therefore not Steam, so I follows. Conclusion II says all Locomotives being Electric is possible. The negative part explicitly supplies a Locomotive outside Electric, so II is impossible. Conclusion III says some Steam being Locomotives is possible. A Steam–Locomotive member can be placed outside Electric and High-Speed without breaking any premise, so III is possible.

The result is I and III. Notice that III is a permission to construct an overlap, not a claim that Steam members have already been proved to exist. This distinction prevents a possibility exercise from being read as an ordinary definite deduction.

Complementary conclusions

Two conclusions can form a complementary pair even when neither follows individually. “Some A are B” and “No A is B” are logical negations: one must be true and the other false. Likewise, “All A are B” and “Some A are not B” are negations. If neither conclusion is independently established, an option saying either member of such a complementary pair holds can be valid.

By contrast, “some A are B” and “some A are not B” can both be true. They are not a complementary pair. “All A are B” and “No A is B” do not generally exhaust the possibilities: a nonempty A may have some members inside B and others outside. Check the actual logical negation rather than relying on one affirmative and one negative sentence.

Conversion and a verification routine

“No A is B” converts to “No B is A.” “Some A are B” converts to “Some B are A.” “All A are B” does not convert to “All B are A,” and it gives “Some B are A” only when A's existence is established. “Some A are not B” does not imply “Some B are not A.”

For each question, translate every premise, mark any known existing member, and test conclusions one by one. For a definite conclusion, try to construct a counterexample. For a possibility, try to construct a supporting model. Finally check that the same diagram satisfies all premises simultaneously. This is more reliable than choosing a conclusion that merely looks natural from a single quick sketch.