# Syllogisms — Solve with Venn Diagrams, Including "Possibility" Questions

Source: https://skills101.in/learn/reasoning/syllogisms — Skills101 free tutorial (Logical Reasoning). Updated 2026-11-09.

A syllogism gives two or three statements (which you must accept as true, even if they sound silly) and asks which conclusions **definitely** follow.

## 1. Draw each statement

![Three Venn diagrams: All A are B (A inside B), Some A are B (overlapping circles), No A is B (separate circles)](https://skills101.in/learn-img/reasoning/syllogism-venn.svg)

| Statement | Drawing | Note |
|---|---|---|
| **All A are B** | A inside B | B may still have things that aren't A |
| **Some A are B** | A and B overlap | At least one A is B. Possibly all! |
| **No A is B** | A and B separate | Nothing in common |
| **Some A are not B** | part of A outside B | |

> **The golden rule:** a conclusion is valid only if it is true in **every** drawing that fits the statements. If you can draw even one picture where it is false, it does **not** follow.

## 2. Solved examples

### Example 1
*Statements: All dogs are animals. All animals are living things.*
*Conclusions: I. All dogs are living things. II. Some living things are dogs.*

Dogs inside animals inside living things. I ✓ (always). II ✓ (the dogs are inside living things, so some living things are dogs). **Both follow.**

### Example 2
*Statements: Some pens are books. Some books are bags.*
*Conclusions: I. Some pens are bags. II. No pen is a bag.*

Draw pens overlapping books, books overlapping bags. Pens and bags **may or may not** touch.
- I does not always hold. II does not always hold.
- But one of them **must** be true (either some pens are bags or none are) → **Either I or II follows.**

> **Either-or** applies when the two conclusions are about the **same pair**, one is "some" and the other "no" (or "all" and "some … not"), and neither follows alone.

### Example 3
*Statements: All cars are vehicles. No vehicle is a toy.*
*Conclusions: I. No car is a toy. II. Some vehicles are cars.*

Cars inside vehicles; toys outside vehicles → toys can't touch cars. I ✓. II ✓ (cars are inside vehicles). **Both follow.**

## 3. Possibility questions

"**Some A being C is a possibility**" asks: can you draw it at all without breaking the statements?

### Example 4
*Statements: All roses are flowers. No flower is a stone.*
*Conclusion: Some roses being stones is a possibility.*

Roses are inside flowers, and flowers can't touch stones → roses can never touch stones → **does not follow**.

### Example 5
*Statements: Some apps are games. All games are free.*
*Conclusion: All apps being free is a possibility.*

Could every app be free? Yes — draw apps inside "free" as well. Nothing forbids it → **follows**.

## Quick rules

- Some + Some → no definite conclusion.
- All + All → All (A ⊂ B ⊂ C gives all A are C).
- All + No → No (if all A are B and no B is C, no A is C).
- "Some" is reversible: some A are B ⇔ some B are A. "All" is **not**: all A are B does not mean all B are A.

## Common mistakes

- Using real-world knowledge. Only the statements count.
- Thinking "some" means "not all". In syllogisms, "some" allows "all".
- Reversing an "all" statement.

## Quick recap

- Draw every statement; test conclusions in **every** valid drawing.
- Some is reversible; all is not.
- Either-or: same pair, complementary conclusions, neither definite alone.
- Possibility: can you draw it without breaking anything?


Practice quiz and free certificate: https://skills101.in/learn/reasoning/syllogisms