60-second revision
- A conclusion follows only if it is true in every Venn diagram that fits the statements.
- A possibility conclusion follows if at least one valid diagram makes it true.
- 'All A are B' converts only to 'Some B are A'; 'Some' and 'No' convert both ways; 'Some A are not B' does not convert.
- All + All = All; All + No = No; Some + All = Some; Some + No = Some not (in that order).
- Two particular statements (Some + Some) give no definite conclusion.
- 'Only a few A are B' = some A are B and some A are not B; 'All A are B' is then impossible.
- Either-or needs a 'some/no' or 'all/some not' pair on the same two terms, with neither definite.
The statement types
| Statement | Venn picture | Converts to |
|---|---|---|
| All A are B | Circle A fully inside circle B | Some B are A |
| Some A are B | A and B overlap | Some B are A |
| No A is B | A and B do not touch | No B is A |
| Some A are not B | Part of A lies outside B | No conversion |
| Only a few A are B | A and B overlap, part of A outside B | Some B are A |
Draw the most restrictive picture you can be sure of, and nothing more. 'Some A are B' shows only a small overlap; it does not say that some A are not B, and it does not rule out all A being B.
Definite conclusions
A definite conclusion follows only if it is true in every diagram that fits all the statements. The quickest test is to draw one diagram for the statements and then try to break the conclusion by moving circles into another legal position. If you can break it, it does not follow.
| First + second (shared middle term) | Conclusion |
|---|---|
| All A are B + All B are C | All A are C |
| All A are B + No B is C | No A is C |
| Some A are B + All B are C | Some A are C |
| Some A are B + No B is C | Some A are not C |
| No A is B + All B are C | Some C are not A |
| All A are B + Some B are C | No definite conclusion |
| Some A are B + Some B are C | No definite conclusion |
Possibility conclusions
Conclusions such as 'All A being C is a possibility' or 'Some A can be C' follow if at least one valid diagram makes them true. They fail only when a definite relation forbids them. Worked example: 'All kites are toys. No toy is a ball.' Can some kites be balls? No — kites sit inside toys and toys never touch balls, so the possibility is ruled out. Can some balls be gifts? Gifts are not mentioned in any statement, so any relation with gifts is possible.
Only a few
'Only a few A are B' is a newer bank-exam form. It means two things at once: some A are B, and some A are not B. So 'Some A are not B' follows directly, and 'All A are B' (even as a possibility) does not follow. Draw it as an overlap with part of A clearly outside B.
The either-or rule
Choose 'either I or II follows' when neither conclusion is definite on its own, both have the same subject and predicate, and the pair is complementary so that one must be true. The usual pairs are 'Some A are B' with 'No A is B', and 'All A are B' with 'Some A are not B'.
Worked example: 'Some caps are hats. Some hats are scarves.' Conclusions: I. Some caps are scarves. II. No cap is a scarf. Neither is definite, but one must hold — either I or II. If instead the statements made 'Some caps are scarves' definite, conclusion II would be false and the either-or option would not apply.
A routine for every question
- Draw the statements one by one in a single Venn diagram, using the minimum overlap.
- For each definite conclusion, try to draw a legal diagram that breaks it. If you can, it does not follow.
- For each possibility conclusion, try to draw a legal diagram that satisfies it. If you can, it follows.
- If both conclusions fail, check whether they form a complementary pair on the same two terms (either-or).
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