Concepts · Syllogisms
A conclusion can sound right and still not follow. Draw it before you decide.
Riya has a syllogism question in a mock test:
All roses are flowers. Some flowers are red. Does “some roses are red” follow?
Roses are flowers, some flowers are red, so some roses are red. It sounds right. She marks “follows” and moves on.
That week the Class 9 students of her school go on a science trip. Everyone in Riya’s tuition group is from the same school. Kabir puts it together: “Then some of your tuition group went.”
“None of them did,” Riya says. “The trip was for Class 9, and we’re all in Class 8.” Then she hears the shape of what he said. All her tuition friends are from her school. Some of her school went on the trip. So some of her tuition friends went. It is her mock question with new words, both statements are true, and the conclusion is false.
So “sounds right” was never a test. The question is whether the statements leave any way for the conclusion to be false. Her tuition group was that way.
Draw each statement as circles. All A are B: A inside B. Some A are B: A and B overlap. No A are B: A and B apart. A conclusion follows only if it is true in every drawing the statements allow.
Some means at least one, and possibly all: the A circle may overlap B, sit inside it, hold B inside itself, or be the same circle. All A are B also allows A and B to be the same circle. Some and No read both ways: some A are B means some B are A, and no A are B means no B are A. All does not (all roses are flowers does not make all flowers roses), and neither does “some A are not B”.
What does follow: some flowers are roses. The roses sit inside the flowers in every drawing.
All cats are pets. No pets are wild.
Draw cats inside pets, and wild apart from pets. However you draw it, the cats are inside the pets, so they never touch wild. “No cats are wild” is true in every drawing: it follows.
Two more kinds of conclusion. One that says “… is a possibility” follows if it is true in at least one drawing: from Riya’s two statements, “all roses being red is a possibility” follows. And when two conclusions about the same two things cannot both be false, as “some A are B” and “no A are B” cannot, but neither follows on its own, the answer is “either I or II follows”.
A syllogism asks whether a conclusion must be true, not whether it could be. A drawing answers that directly: if you can draw one picture where the statements are true and the conclusion is false, it does not follow.
Counting the regions of circles is a topic of its own, Venn diagrams.
1. Statements: All pens are blue. All blue things are bright.
Conclusions: I. All pens are bright. II. All bright things are pens.
Draw pens inside blue, and blue inside bright. Could there be a bright thing outside the pens?
Only I follows
Pens sit inside blue, and blue inside bright, so pens are inside bright in every drawing: I follows.
Bright can be much bigger than pens, so II does not follow. All never reads backwards.
2. Statements: Some boys are singers. No singer is a dancer.
Conclusions: I. Some boys are not dancers. II. Some singers are boys.
Look at the boys who are singers. Where can they be?
Both follow
The boys who are singers are inside the singers circle, which never touches dancers, so those boys are not dancers: I follows.
Some reads both ways: some boys are singers means some singers are boys. II follows too.
3. Statements: All cups are mugs. Some mugs are red.
Conclusions: I. Some cups are red. II. All mugs are cups.
Draw cups inside mugs. Where can the red part of the mugs go?
Neither follows
The red part of the mugs can miss the cups altogether, so I does not follow.
All never reads backwards: all cups are mugs does not make all mugs cups. Neither follows.
A week later:
Some pencils are pens. All pens are books. Does “some pencils are books” follow?
Riya draws pens inside books, and pencils overlapping pens. The pencils that are pens sit inside books in every drawing. She marks “follows”, and this time she has drawn it.
Every question in the app is answered before its solution shows.
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