The four statement types
| Type | Example | Venn picture |
|---|---|---|
| All A are B | All pens are tools | A circle inside B |
| Some A are B | Some tools are metal | The circles overlap |
| No A is B | No pen is a bird | Separate circles |
| Some A are not B | Some tools are not metal | Part of A outside B |
Useful conversions: "Some A are B" can be read as "Some B are A"; "No A is B" can be read as "No B is A"; "All A are B" does not mean "All B are A" (only "Some B are A" follows, provided A exists).
A method
- Draw the statements as Venn circles, using the most definite diagram for each.
- Draw all the possible diagrams that fit, because "some" statements allow several.
- Test each conclusion against every diagram.
- A conclusion follows only if it holds in every possible diagram.
- "Possibility" conclusions follow if they hold in at least one diagram and are not ruled out.
Worked examples
Example 1. Statements: All engines are machines. Some machines are vehicles. Conclusions: (I) Some engines are vehicles. (II) Some vehicles are machines. Draw engines inside machines; vehicles overlap machines, but they may overlap the engines part or only the rest of machines. (I) need not hold, because the vehicles may lie in the part of machines that is outside engines. (II) follows from the second statement. Only conclusion II follows.
Example 2. Statements: No pen is a book. All books are objects. Conclusion: Some objects are not pens. Books are objects and no book is a pen; so the books are objects that are not pens. Hence "Some objects are not pens" follows.
Example 3. Statements: Some cats are dogs. All dogs are animals. Conclusions: (I) Some cats are animals. (II) All cats are animals. The cats that are dogs are animals, so (I) follows. (II) does not follow, because some cats may not be dogs.
Example 4 (possibility). Statements: All roses are flowers. Some flowers are red. Conclusion: Some roses may be red. The red flowers could include roses; nothing rules this out; so the possibility follows.
Example 5 (either-or). Statements: Some A are B. Conclusions: (I) All A are B. (II) No A is B. "Some A are B" rules out (II), because at least one A is a B, but it does not establish (I), since some A may not be B. So neither follows. An "either-or" option applies only when the two conclusions are complementary, so that one of them must be true; that is not the case here.
Common traps
- Assuming "some" means "some but not all". In logic, "some" means at least one, and may include all.
- Assuming "all A are B" implies "all B are A".
- Using real-world knowledge. Judge only from the statements.
- Drawing only one diagram.
A practice routine
- Day 1: the four statement types and their conversions.
- Day 2: two-statement problems with Venn diagrams.
- Day 3: three-statement problems and possibility conclusions.
- Day 4: a timed set of 12 at 50 seconds each.
Where it appears
Syllogisms appear in banking reasoning, SSC CGL reasoning, railway reasoning, MAT, CLAT Logical Reasoning and XAT Decision Making analytical sets. Related: blood relations and puzzles.
Practice tests below are published by institutes and educators on TestMint.
