Syllogisms Exit
Logical reasoning

Syllogisms

Draw circles (Venn), never reason in words. A conclusion follows only if it is true in EVERY possible drawing.

Start with this

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.

AB
All A are B: the A circle sits inside the B circle
AB
No A is B: the circles do not touch
AB
Some A are B: the circles overlap (at least one A is a B)
AB✗
Some A are not B: part of A lies outside B

Worked example

Statements: All cats are animals. Some animals are black. Conclusion: Some cats are black. Does it follow?

  1. Draw cats inside animals.
    Why: First statement.
  2. Draw 'black' overlapping animals, but outside the cat circle.
    Why: Allowed: 'some animals are black' does not say which animals.
  3. In that picture no cat is black, so the conclusion does not follow.

Answer: 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').

Worked example

Statements: Some pens are books. No book is a bag. Conclusion: Some pens are not bags. Does it follow?

  1. The pens that are books cannot be bags (no book is a bag).
    Why: Follow the overlap.
  2. So at least those pens are not bags. It follows.

Answer: Follows

Shortcut: Break the conclusion

Use it when: To test a conclusion, try to draw a diagram where it is FALSE.

  1. If you can, it does not follow.
  2. If you cannot, it follows.

Break the conclusion: try it

All cats are dogs. Some dogs are birds. Conclusion: some cats are birds?

  1. Draw birds overlapping dogs but not the cat circle.
  2. Possible, so it does not follow.

Answer: Does not follow

Your turn medium

All boys are girls. No girl is a teacher. Does 'No boy is a teacher' follow?

  1. All + No = No.

Answer: Follows

Your turn medium

Some apples are mangoes. Some mangoes are grapes. Does 'Some apples are grapes' follow?

  1. Some + Some = no definite conclusion.

Answer: Does not follow

Your turn hard

No car is a bus. All buses are trucks. Conclusions: I. Some trucks are not cars. II. No truck is a car.

  1. The buses inside trucks are not cars, so some trucks are not cars.
  2. Other trucks may be cars, so II fails.

Answer: Only I follows

Recap

Now practise

5 questions with full solutions.

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