Last reviewed 22 Sept 2026 · 5 min read
The four forms
| Form | Statement | What it allows |
|---|---|---|
| A (universal affirmative) | All A are B | every A is inside B; B may be bigger |
| E (universal negative) | No A is B | the circles do not touch |
| I (particular affirmative) | Some A are B | the circles overlap; they may also coincide |
| O (particular negative) | Some A are not B | part of A lies outside B |
A conclusion is valid only if it holds in every diagram the statements allow. One legal diagram in which it fails is enough to reject it. So the work is not to draw a diagram, but to draw the diagram that commits to the least — and then to try to break the conclusion.
Drawing the least-committed diagram
- All A are B: A wholly inside B. (Do not make them equal unless forced.)
- No A is B: two separate circles.
- Some A are B: two overlapping circles — and remember that "some" does not deny "all".
- Some A are not B: A overlapping B with a part of A clearly outside.
Then read the conclusion and ask: can I redraw this, still obeying the statements, so that the conclusion becomes false? If yes, it does not follow.
Conversions that are always safe
- All A are B → Some B are A (and Some A are B).
- No A is B → No B is A, and Some A are not B.
- Some A are B → Some B are A.
- Some A are not B → nothing about B and A in reverse.
Possibility conclusions
Modern papers ask about possibility: "All A being C is a possibility." Such a conclusion is true when at least one legal diagram contains it — the mirror image of the main rule. So:
- A definite conclusion needs every diagram.
- A possibility conclusion needs one diagram.
Either-or (complementary pairs)
When neither conclusion follows alone, but together they cover every case and overlap in none, the answer is either I or II follows. The pair must:
- have the same subject and predicate, and
- be of the forms (Some A are B, No A is B) or (All A are B, Some A are not B).
Worked examples
Statements: All pens are books. All books are tables. Conclusions: I. All pens are tables. II. Some tables are pens.
Solution. Pens inside books inside tables, in every legal drawing → I follows. From "all pens are tables", some tables are pens → II follows. Both follow.
Statements: Some cats are dogs. Some dogs are rats. Conclusions: I. Some cats are rats. II. No cat is a rat.
Solution. Cats and rats may overlap, or may not — both drawings are legal, so neither conclusion follows on its own. But the two together cover every case and overlap in none → either I or II follows.
Statements: All fruits are sweet. No sweet thing is bitter. Conclusions: I. No fruit is bitter. II. Some bitter things are fruits.
Solution. Fruits lie inside sweet, which is separate from bitter → I follows in every drawing; II contradicts it → only I follows.
Statements: Some books are pens. All pens are red. Conclusions: I. Some books are red. II. All books are red.
Solution. The books that are pens are red → I follows. II fails in the drawing where other books lie outside red → only I follows.
Statements: All roses are flowers. Some flowers are red. Conclusion: All roses being red is a possibility.
Solution. A legal drawing exists with every rose inside the red region → the possibility holds.
Statement: Only boys can play. Conclusion: All who play are boys.
Solution. "Only A are B" means all B are A → the conclusion follows. Note it does not mean that all boys play.
Statements: All A are B. No B is C. Some C are D. Conclusions: I. No A is C. II. Some D are not B.
Solution. A inside B, and B separate from C → I follows. The C's that are D lie outside B, so some D are not B → II follows. Both follow.
Which set of statements makes "Some pens are papers" definitely true? (a) All pens are books, all books are papers (b) Some pens are books, some books are papers
Solution. (a) — pens inside books inside papers forces the overlap. In (b) the middle term connects nothing.