Syllogism made clear.
Judge only what must follow from the statements. A conclusion that sounds realistic is not enough: it must remain true in every arrangement allowed by the premises.
Key concepts in Syllogism
All A are B
A lies inside B. This does not imply that all B are A.
No A are B
A and B do not overlap. This relation can be reversed: no B are A.
Some A are B
At least one member belongs to both groups. This implies some B are A.
Some A are not B
At least one A lies outside B. It says nothing definite about every other A.
Existence convention
Here universal statements alone do not guarantee that a group has members. “Some” explicitly guarantees existence.
Read the set relationship
| Statement | Visual meaning | Safe inference |
|---|---|---|
| All A are B | A ⊆ B | Not necessarily B ⊆ A |
| No A are B | A ∩ B is empty | No B are A |
| Some A are B | A ∩ B has a member | Some B are A |
A reliable solving method
- Represent each category with a region or a set relationship.
- Place only overlaps or exclusions that the premises require.
- Try to construct a counterexample to each conclusion.
- Accept a conclusion only if no permitted counterexample exists.
Step-by-step solved examples
All roses are flowers. All flowers are plants. Must all roses be plants?
- Roses are a subset of flowers.
- Flowers are a subset of plants.
- Therefore roses are a subset of plants: the conclusion follows.
Some books are gifts. All gifts are useful. Must some books be useful?
- The first statement guarantees at least one book that is a gift.
- Every gift belongs to the useful group.
- That same book is useful, so the conclusion follows.
All cats are animals. Must all animals be cats?
- The statement puts cats inside animals.
- Animals may include members outside the cat group.
- The reversed universal claim does not follow.
Syllogism: practice questions
Choose one answer before opening the explanation. Write down your reasoning as well as the option; a correct guess is different from a method you can repeat.
All A are B; all B are C. Which follows?
Show answer and explanation — question 1
Answer A: All A are C
Subset inclusion is transitive: A lies inside C.
No pens are cups. Which follows?
Show answer and explanation — question 2
Answer B: No cups are pens
Non-overlap is symmetric.
Some A are B. Which follows?
Show answer and explanation — question 3
Answer C: Some B are A
The same overlap member is both B and A.
All A are B, no B are C. Which follows?
Show answer and explanation — question 4
Answer D: No A are C
Anything in A is in B and cannot be in C.
Some A are B; some B are C. Must some A be C?
Show answer and explanation — question 5
Answer B: No, it is not forced
The two “some” groups can be different members of B.
Under this lesson’s convention, “All A are B” alone implies some A exist?
Show answer and explanation — question 6
Answer C: No
A universal statement does not itself assert existence.
Frequently asked questions
What does “does not follow” mean?
It means the conclusion is not guaranteed, not necessarily that it is false.
Can outside knowledge help?
No. Treat the categories as labels and use only the supplied premises.
Represent each category with a region or a set relationship. Accept a conclusion only if no permitted counterexample exists.