Syllogisms
Draw circles (Venn), never reason in words. A conclusion follows only if it is true in EVERY possible drawing.
Why this topic matters
All cats are animals. Some animals are black. Does it follow that some cats are black? No. Draw the circles and you can see why.
Syllogisms are decided by pictures, not by common sense. The statements may be silly ('All pens are dogs'); treat them as true and draw.
Four kinds of statement, four pictures
All A are B: the A circle is inside B. No A is B: the circles do not touch. Some A are B: the circles overlap. Some A are not B: part of A is outside B.
A conclusion follows only if it is true in EVERY way you could draw the circles. If you can draw one picture where it is false, it does not follow.
- Draw cats inside animals.First statement.
- Draw 'black' overlapping animals, but outside the cat circle.Allowed: 'some animals are black' does not say which animals.
- In that picture no cat is black, so the conclusion does not follow.
'Either-or' and possibility
If two conclusions are 'Some A are B' and 'No A is B', one of them must be true even when neither follows alone. The answer is 'either I or II follows'.
'Some A being B is a possibility' follows whenever the circles are allowed to overlap (nothing says 'No A is B').
- The pens that are books cannot be bags (no book is a bag).Follow the overlap.
- So at least those pens are not bags. It follows.
Formula summary
Shortcuts and tricks
Break the conclusion
Use it when: To test a conclusion, try to draw a diagram where it is FALSE.
- If you can, it does not follow.
- If you cannot, it follows.
- Draw birds overlapping dogs but not the cat circle.
- Possible, so it does not follow.
Before you move on, you should be able to...
- draw the four statement types
- accept a conclusion only if no picture breaks it
- spot either-or pairs
Practice questions (5)
Try each one before opening the solution. Or practise them one by one so your score is saved.
easy All pens are books. All books are tables. Does 'All pens are tables' follow?
- All + All = All.
Answer: Follows
easy Some roses are flowers. All flowers are plants. Does 'Some roses are plants' follow?
- Some + All = Some.
Answer: Follows
medium All boys are girls. No girl is a teacher. Does 'No boy is a teacher' follow?
- All + No = No.
Answer: Follows
medium Some apples are mangoes. Some mangoes are grapes. Does 'Some apples are grapes' follow?
- Some + Some = no definite conclusion.
Answer: Does not follow
hard No car is a bus. All buses are trucks. Conclusions: I. Some trucks are not cars. II. No truck is a car.
- The buses inside trucks are not cars, so some trucks are not cars.
- Other trucks may be cars, so II fails.
Answer: Only I follows
Class plan (2 hours, for trainers)
A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.
| 10 min | Warm-up. |
| 30 min | Four pictures; draw each statement. |
| 25 min | Testing conclusions: find one picture that breaks it. |
| 15 min | Either-or and possibility. |
| 35 min | Practice set. |
| 5 min | Recap. |