Syllogism: Statements and Conclusions

What does the question give you?

You get two or three statements about groups of things. Then you get two conclusions.

You must pick the conclusions that follow from the statements.

What is the one big rule?

Treat every statement as true, even if it sounds silly in real life.

Never use outside knowledge. Only the statements count.

What does “follows” mean?

A conclusion follows only if it is true in every possible picture of the statements.

If it is true in one picture and false in another, it does not follow.

What are the four types?

All A are B. No A is B. Some A are B. Some A are not B.

The first two are called universal. The last two are called particular.

What does each one mean?

All A are B means every A is inside B. No A is B means the two groups never meet.

Some A are B means at least one A is also B. Some A are not B means at least one A is outside B.

Does some mean not all?

No. In these questions, the word some means at least one. It may even be all.

So some A are B does not say that some A are not B.

Which statements flip safely?

No A is B flips to No B is A. Some A are B flips to Some B are A.

In both, the two groups touch in the same way from either side.

What happens to “All A are B”?

It flips only to Some B are A. It does not flip to All B are A.

Think of dogs and animals. All dogs are animals, but not all animals are dogs.

What about “Some A are not B”?

It does not flip at all. Some students are not boys does not tell you about boys.

Leave it as it is.

How do you draw the statements?

Draw one circle for each group. Put the name of the group on the circle.

All A are B means circle A sits fully inside circle B.

How do you draw No and Some?

No A is B means two separate circles. Some A are B means two circles that overlap.

Mark the overlap with a small dot if you know something lives there.

What if the picture can change?

Try to draw a second picture that still obeys the statements.

If a conclusion breaks in any picture, it does not follow.

Can you see why the reverse fails?

All dogs are animals. Draw the dog circle inside the animal circle.

A cat lives in the animal circle but outside the dog circle. So All animals are dogs is false.

Why this works: the big circle has room left over that the small circle does not fill.

Example one: a chain of Alls

Statements: All pens are tools. All tools are metals. Conclusion: All pens are metals.

Pens sit inside tools, and tools sit inside metals. So pens sit inside metals. The conclusion follows.

Why this works: each All statement puts one circle inside the next, so the smallest circle is inside the biggest.

Example two: Some and No

Statements: Some doctors are singers. No singer is a pilot. Conclusion: Some doctors are not pilots.

Take the doctors who are singers. None of them can be pilots. So the conclusion follows.

Why this works: the doctors who are singers all lie outside the pilot circle. So at least one doctor is not a pilot.

Example three: Some and Some

Statements: Some roses are flowers. Some flowers are red. Conclusion: Some roses are red.

The roses and the red flowers may be different flowers. So the conclusion does not follow.

Why this works: one picture, where the roses are outside the red part, breaks the conclusion. One breaking picture is enough to reject it.

Try it: draw roses overlapping flowers on the left, and red overlapping flowers on the right, with no overlap between roses and red.

Which pairs give a conclusion?

All + All gives All. All + No gives No. Some + All gives Some. Some + No gives Some not.

In each pair, the middle term must be shared by both statements.

Which pairs give nothing?

Some + Some gives no sure result. No + No gives no sure result.

Two statements with the shared term at the end of both also give no sure result.

What is a complementary pair?

Sometimes conclusion one says Some A are B and conclusion two says No A is B.

Neither is sure, but one of them must be true. Then the answer is Either I or II follows.

Can you see All + No solved?

Statements: All cats are pets. No pet is a stone. Conclusion: No cat is a stone.

Cats sit inside pets. Pets never meet stones. So cats never meet stones. The conclusion follows.

Why this works: whatever is true of the whole group is true of the smaller group inside it.

Can you see an either-or case?

Statement: Some kites are red. Conclusion I: Some kites are not red. Conclusion II: All kites are red.

Neither is sure, because the statement allows both pictures. But one of them must be true.

So the answer is Either I or II follows. Why this works: the two conclusions cover every possible picture between them.

What mistakes cost marks?

Flipping All A are B into All B are A. Reading some as not all.

Using real-world knowledge about the words in the question.

How do you stay quick?

Redraw every statement as a circle. Check each conclusion against your drawing.

With three statements, join them two at a time.

How should you practise?

Solve five questions a day. Draw circles for each one.

Then try the quiz below and read every explanation.