Syllogism items are the fairest questions on the paper: two or three statements, one or two claimed conclusions, and a purely formal question — does the conclusion follow from the statements as written, with no outside knowledge? You do not need to know anything about the subject matter, only the rules of the form.
They are also where candidates lose marks to common sense. "All cats are animals; some animals are black; therefore some cats are black" feels true because you know cats can be black, yet it does not follow: the black animals might all be dogs. The exam tests the form, not the world.
Turn the statements into circles and ask one question: is the conclusion true in every arrangement the statements allow?
| Statement | Diagram |
|---|---|
| All A are B | The A circle sits entirely inside B |
| No A are B | The circles do not touch |
| Some A are B | The circles overlap, with at least one item inside the overlap |
| Some A are not B | Part of A lies outside B |
The counterexample method is faster than memorising rules: try to draw a picture consistent with the statements in which the conclusion is false. If the drawing is possible, the conclusion does not follow. Note the exam convention: some means at least one, and possibly all.
| Premises | Conclusion | Why |
|---|---|---|
| All A are B; All B are C | All A are C | Nested circles chain |
| All A are B; No B are C | No A are C | A lies inside B, which avoids C |
| Some A are B; All B are C | Some A are C | The overlap item is also in C |
| No A are B | No B are A | Non-contact is symmetric |
| Some A are B | Some B are A | Overlap is symmetric |
| All A are B | Some B are A | Only valid if A exists — the exam accepts it |
| Premises | Claimed conclusion | Counterexample |
|---|---|---|
| All A are B | All B are A | Put B wider than A: every A is B, most B are not A |
| Some A are B; Some B are C | Some A are C | A and C can overlap B at different places, or not meet |
| All A are B; Some C are B | Some A are C | C can overlap B entirely outside A |
| No A are B; Some C are A | No C are B | C can extend beyond A into B |
Valid. Engineer circle inside graduate circle, graduate circle inside literate circle, so engineer is inside literate. This is the chaining form.
Valid. The doctors who are musicians are also artists, and there is at least one such doctor by "some".
Invalid. Draw the part of the flower circle that lies outside the rose circle and let all the quickly-fading flowers be there. The premises remain true and the conclusion is false, so it does not follow.
Valid. The snake circle is inside the reptile circle, and the reptile circle avoids mammals entirely, so snakes cannot be mammals.
All engineers are graduates. All graduates are literate. Which conclusion follows?
Some doctors are musicians. All musicians are artists. Which conclusion follows?
All roses are flowers. Some flowers fade quickly. Which conclusion follows?
Some A are B. Some B are C. Which conclusion follows?
Without looking back: which two particular premises never produce a conclusion, and what does "some" mean in these questions?
The answer: "some A are B" with "some B are C" yields nothing; and "some" means at least one, possibly all.
In these questions "some" means at least , and possibly all. "All A are B" and "All B are C" together give all A are . To disprove a claimed conclusion you draw a , and outside knowledge must be .
Statements: "No student who arrived late was admitted." "All those admitted sat the test." Which of these follow: (1) No student who arrived late sat the test; (2) All who sat the test were admitted; (3) Some who sat the test were admitted?
Let A be latecomers, B the admitted and C those who sat the test. The premises are: No A are B; All B are C.
(1) "No A are C" does not follow: the admitted circle sits inside the test circle, but a latecomer could sit the test without being admitted — the test circle is wider than the admitted circle, and latecomers may fall in that outer ring. Invalid.
(2) "All C are B" is the converse of the second premise, so it does not follow: many who sat the test may not have been admitted. Invalid.
(3) "Some C are B" follows, provided at least one student was admitted: with all admitted sitting the test, those admitted students are inside both circles. Valid on the usual exam convention that the described group exists.
This item shows why naming the three circles and their relative sizes answers the question faster than reading the sentences again — the whole problem is the shape, not the story.
Formal logic is done. The next lesson returns to puzzles about people, but with a map instead of a grid: family relationships and directions, where every sentence adds an arrow or a branch.