Syllogism: Venn Diagram Method

A syllogism question gives two or three statements and asks which conclusions must follow from them. Every statement is one of four types, made from two choices: is it about all members or only some, and is it affirmative or negative? That gives 2 × 2 = 4 types.

Statement	Meaning in the diagram
All A are B	The A circle lies wholly inside B
No A is B	A and B do not touch
Some A are B	A and B overlap in at least one point
Some A are not B	Part of A lies outside B

Two rules apply throughout. Treat every statement as true, however odd it sounds. And assume each term has at least one member, which is the usual convention in exams.

The syllogism Venn diagram method works like this: draw only what the statements force, and nothing more. Any overlap the statements do not require is left out of the basic diagram, or drawn as a dotted possibility. This basic picture is the minimum-overlap diagram.

With two terms there are 2 × 2 = 4 regions: in both, in A only, in B only, and in neither.

With three terms there are 2³ = 8 regions. The statements decide which of them must be filled and which must be empty; the rest are unknown.

A conclusion follows only if it is true in every diagram the statements allow. If you can draw even one allowed diagram where it fails, it does not follow.

Worked example 1: a chain of 'all'

All dogs are animals. All animals are living. Draw dogs inside animals, and animals inside living things. I: All dogs are living, which follows. II: Some living things are dogs, which also follows, because dogs exist and all of them are inside the living circle. Both follow.

Worked example 2: 'some' then 'all'

Some pens are books. All books are bags. The pens that are books are also bags, so I: Some pens are bags follows. II: All bags are pens does not, because the bag circle can extend far beyond the pens. Only I follows.

Worked example 3: a negative statement

No cat is a rat. All rats are mice. I: No cat is a mouse does not follow, because the cat circle may still touch the part of the mouse circle outside the rats. II: Some mice are not cats follows, because the rat part of the mice can never be cats. Only II follows.

Sometimes neither conclusion follows on its own, but together they cover every case. This happens when the two conclusions are about the same two terms and form a complementary pair:

"Some A are B" with "No A is B"
"All A are B" with "Some A are not B"

In any diagram exactly one of the pair is true. So if neither follows alone, the answer is "either I or II follows".

Worked example 4: either-or

Some A are B. Some B are C. A and C may or may not overlap, so neither "Some A are C" nor "No A is C" is certain. They form a complementary pair, so either I or II follows.

Check the pair carefully. "Some A are C" and "No C is A" also qualify, because "no" statements work both ways. But "Some A are C" and "Some A are not C" do not form a pair, since both can be true at once.

Newer question styles ask whether something "is a possibility". The test is reversed: a possibility conclusion holds if you can draw at least one allowed diagram where it is true. It fails only if the statements make it impossible.

Worked example 5: possible or not

All roses are flowers. No flower is a tree. I: All trees being roses is a possibility. This is impossible, because trees can never touch flowers, so they can never be roses. II: Some flowers not being roses is a possibility. Yes; the flower circle can be larger than the rose circle. Only II follows.

Draw the minimum-overlap diagram from the statements.
For each definite conclusion, try to draw an allowed diagram that breaks it.
For each possibility conclusion, try to draw an allowed diagram that makes it true.
If neither definite conclusion follows, check whether they form an either-or pair.

With three terms, a conclusion can link any 2 of them, giving 3 possible pairs of terms. Conclusions linking the two terms that never appear in the same statement are the ones that most often fail.

With four terms across three statements, there are 6 possible pairs, so draw the full chain before reading the conclusions.

✘ Using real-world knowledge instead of the statements.
✘ Reversing "All A are B" into "All B are A".
✘ Marking "either-or" for two conclusions about different terms.
✘ Treating a possibility as definite, or the reverse.

Careful, step-by-step diagrams also help with blood relations and order and ranking. For more reasoning methods, see the study hub.

Q1. Statements: All pencils are pens. Some pens are erasers. Conclusions: I. Some pencils are erasers. II. Some pens are pencils. easy

(a) Only II follows (b) Only I follows (c) Both follow (d) Neither follows

✅ Answer: (a) Only II follows

💡 Explanation: The erasers may overlap the pens only outside the pencil circle, so I is not certain. Every pencil is a pen and pencils exist, so some pens are pencils: II follows.

Q2. Statements: No book is a copy. All copies are files. Conclusions: I. Some files are not books. II. No file is a book. medium

(a) Neither follows (b) Only I follows (c) Only II follows (d) Both follow

✅ Answer: (b) Only I follows

💡 Explanation: The files that are copies can never be books, so some files are not books: I follows. Other files outside the copy circle may still be books, so II is not certain.

Q3. Statements: All A are B. All B are C. Conclusions: I. All A are C. II. Some C are A. easy

(a) Only I follows (b) Only II follows (c) Both follow (d) Neither follows

✅ Answer: (c) Both follow

💡 Explanation: A lies inside B and B lies inside C, so all A are C. Since A is not empty, the A part of C means some C are A. Both follow.

Q4. Statements: Some cars are buses. No bus is a train. Conclusions: I. Some cars are not trains. II. Some trains are not buses. medium

(a) Only I follows (b) Only II follows (c) Neither follows (d) Both follow

✅ Answer: (d) Both follow

💡 Explanation: The cars that are buses cannot be trains, so I follows. No train is a bus at all, and trains exist, so some trains are not buses: II follows too.

Q5. Statements: Some X are Y. No Y is Z. Conclusions: I. Some X are Z. II. No X is Z. hard

(a) Only I follows (b) Either I or II follows (c) Neither follows (d) Only II follows

✅ Answer: (b) Either I or II follows

💡 Explanation: X and Z may or may not overlap, so neither conclusion is certain alone. They share the same terms and one must be true, a 'some' and 'no' pair, so either I or II follows.

Q6. Statements: All teachers are graduates. Some graduates are doctors. Conclusions: I. Some teachers being doctors is a possibility. II. No graduate being a teacher is a possibility. hard

(a) Both follow (b) Only II follows (c) Only I follows (d) Neither follows

✅ Answer: (c) Only I follows

💡 Explanation: A diagram where the doctor circle reaches into the teachers is allowed, so I is possible. Teachers exist and all are graduates, so 'no graduate is a teacher' can never be true: II fails.

Does 'some' include the possibility of 'all'?

Yes. In syllogism, 'some A are B' means at least one A is B, and it does not rule out all A being B. So a diagram where A lies wholly inside B is still allowed by a 'some' statement.

Can I use real-world knowledge to judge a conclusion?

No. Treat every statement as true, even if it sounds absurd, such as 'all cats are chairs'. Judge conclusions only from the statements given. Real-world facts are a common trap built into these questions.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

How this page was prepared. Drafted with AI assistance (Claude (Anthropic AI model)) from the sources listed above. 5 calculations on this page were recomputed by software and matched. Automated checks also covered structure, internal links and duplicate text. Software checks cannot confirm facts. Reviewed by praveenkumar on 2026-10-20 against the sources listed above.

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