Syllogisms Solved with Venn Diagrams

Syllogism questions give statements and ask which conclusions follow. Drawing circles (Venn diagrams) turns confusing wording into a picture.

The four statement types

Statement	Meaning	Diagram
All A are B	every A is inside B	circle A entirely inside circle B
No A is B	A and B don't overlap	two separate circles
Some A are B	at least some A are B	circles overlap
Some A are not B	at least some A lie outside B	part of A outside B

The key rule

A conclusion is correct only if it is true in every possible diagram. If the conclusion holds in one diagram but not another, it doesn't follow.

Example 1

Statements: All cats are animals. All animals are living beings. Conclusion: All cats are living beings.

Circle "cats" inside "animals", which is inside "living beings". So cats are inside living beings. The conclusion follows.

Example 2

Statements: All dogs are pets. Some pets are cats. Conclusion: Some dogs are cats.

The pets circle contains dogs. The cats circle overlaps pets, but might not overlap the dogs part. The conclusion does not follow.

Example 3

Statements: Some pens are books. All books are papers. Conclusion: Some pens are papers.

Books are inside papers, and some pens overlap books, so those pens are in papers. The conclusion follows.

Common shortcuts

Premises	Conclusion
All A are B; all B are C	All A are C
All A are B; some B are C	no definite conclusion
Some A are B; all B are C	Some A are C
No A is B; all C are A	No C is B
All A are B; no B is C	No A is C

Possibility

Some questions use "possibility" conclusions, such as "Some A may be B". These are true if the diagram can be drawn that way, even if it isn't forced.

Either-or

Some exams include an "either-or" option when two conclusions are complementary, such as "Some A are B" and "No A is B", and neither follows alone. Check the instructions of your exam.

Tips

1. Draw the most general diagrams, allowing all possibilities.
2. Test each conclusion against every diagram.
3. Ignore real-world facts. Use only the statements given.

Practice

Syllogisms improve with repetition. Ten a day soon builds confidence.

Practise with Online Exam Question & Answer →