Syllogisms

Syllogism problems give us 2-3 statements (premises) and ask us to figure out which conclusions are valid. The secret weapon is Venn diagrams — once we draw the circles correctly, the answer is staring at us. No need to “think” logically in our heads; just draw and read.

The Four Statement Types

Every syllogism statement falls into one of four types. Let’s learn them with their Venn diagram representations.

Venn Diagram Representations

The Method: Draw and Check

Step 1: Draw Venn diagrams for the given statements (premises).

Step 2: Check if each conclusion is necessarily true in EVERY possible diagram.

Step 3: A conclusion is definite only if it’s true in ALL valid diagrams. If it fails in even one valid arrangement, it’s not a definite conclusion.

”Definite” vs “Possibility”

• Definite conclusion: “Some A are B” — MUST be true based on the premises
• Possibility conclusion: “Some A are B is a possibility” — CAN be true (not necessarily true, but not contradicted either)

Worked Examples

Example 1: Basic two-statement syllogism

Statements:

1. All dogs are animals
2. All animals are living things

Conclusions:

• I. All dogs are living things
• II. Some living things are dogs

Draw the Venn diagram: Dogs inside Animals, Animals inside Living Things. So Dogs is inside Living Things too.

Conclusion I: All dogs are living things → Dogs circle is fully inside Living Things. TRUE ✓

Conclusion II: Some living things are dogs → Since all dogs are living things, there’s an overlap (the dogs themselves). TRUE ✓

Both conclusions follow.

Example 2: With “No” statement

Statements:

1. All cats are pets
2. No pets are wild

Conclusions:

• I. No cats are wild
• II. Some wild are cats

Venn diagram: Cats inside Pets. Wild completely separate from Pets. Since Cats is inside Pets and Wild doesn’t touch Pets, Cats and Wild are completely separate too.

Conclusion I: No cats are wild → Cats and Wild are separate. TRUE ✓

Conclusion II: Some wild are cats → This needs overlap between Wild and Cats. But they’re completely separate. FALSE ✗

Only conclusion I follows.

Example 3: The “Some” trap

Statements:

1. Some roses are flowers
2. All flowers are beautiful

Conclusions:

• I. All roses are beautiful
• II. Some roses are beautiful

Draw: Roses and Flowers overlap. Flowers inside Beautiful. The overlapping part of Roses-and-Flowers is inside Beautiful. But the part of Roses outside Flowers may or may not be inside Beautiful.

Conclusion I: All roses are beautiful → Not necessarily. Only the roses that are flowers are definitely beautiful. The roses outside the Flowers circle might not be beautiful. NOT DEFINITE ✗

Conclusion II: Some roses are beautiful → The roses that ARE flowers are inside Beautiful. So yes, at least some roses are beautiful. TRUE ✓

Only conclusion II follows.

Example 4: Possibility-type question

Statements:

1. All apples are fruits
2. Some fruits are sweet

Conclusions:

• I. Some apples are sweet
• II. “Some apples are sweet” is a possibility

For Conclusion I: Apples is inside Fruits. Some Fruits are sweet (overlap between Fruits and Sweet). But the sweet part might not overlap with the Apple part — it could be entirely in the non-Apple part of Fruits. So “Some apples are sweet” is NOT definite.

For Conclusion II: CAN some apples be sweet? Yes — it’s possible that the sweet part of Fruits includes some apples. Nothing prevents it. So it IS a possibility. ✓

Example 5: Three-statement syllogism

Statements:

1. All A are B
2. Some B are C
3. No C are D

Conclusions:

• I. Some A are C
• II. No A are D

Venn diagram: A inside B. B and C overlap. C and D are separate.

Conclusion I: Some A are C → The B-C overlap might or might not include the A part of B. So this is not definite. ✗

Conclusion II: No A are D → Even if some A were C, those wouldn’t be D (since no C are D). But A could potentially be D through parts of B that aren’t C. Actually, A is inside B, and some B are C but not all B are C. The non-C parts of B could potentially overlap with D (nothing prevents it). So “No A are D” is not definite either. ✗

Neither conclusion follows.

The Complementary Pair Rule

This is a powerful shortcut that can save time on “either…or” type questions.

In simple language: for any two groups A and B, either they overlap (Some A are B) or they don’t (No A are B). There’s no third option. Similarly, either all of A is inside B (All A are B) or some part of A is outside B (Some A are not B).

Example using the complementary pair:

Statements: Some pens are pencils. All pencils are erasers.

Conclusions:

• I. Some pens are erasers
• II. No pens are erasers

Conclusion I: Some pens are pencils, all pencils are erasers → the pens that are pencils are also erasers. So “Some pens are erasers” is TRUE.

Since Conclusion I is true, we don’t even need the complementary pair rule. But if neither had been individually provable and the pair was “Some A are B” and “No A are B,” we’d know one must be true.

Common Traps

1. “All A are B” does NOT mean “All B are A.” If all dogs are animals, that doesn’t mean all animals are dogs.

2. “Some” means “at least one.” “Some A are B” could mean ALL A are B. “Some” doesn’t imply “not all.”

3. Don’t use real-world knowledge. If the premise says “All fish are birds,” accept it. Don’t think “but fish aren’t birds!” In syllogisms, only the stated premises matter.

4. Drawing only one Venn diagram isn’t enough. We need to check if a conclusion holds in ALL possible valid diagrams. If we can draw even one valid diagram where the conclusion fails, it’s not a definite conclusion.

Quick Method Summary

1. Draw the Venn diagram based on premises
2. For “definite” conclusions: must be true in ALL valid diagrams
3. For “possibility” conclusions: must be true in AT LEAST ONE valid diagram
4. Check complementary pairs when individual conclusions don’t follow
5. Ignore real-world knowledge — only use the given statements

Common Exam Variations

• Two statements, two conclusions — most common format
• Three statements, multiple conclusions — harder, more circles
• “Either I or II follows” — usually involves complementary pairs
• Possibility-based conclusions — “Some A are B is a possibility”
• Negative premises — “No A are B” combined with “All B are C”

Practice Problems

Problem 1: Statements: All books are pens. Some pens are erasers. Conclusions: I. Some books are erasers. II. Some erasers are pens.

Problem 2: Statements: No cat is a dog. All dogs are rats. Conclusions: I. No cat is a rat. II. Some rats are dogs.

Problem 3: Statements: All A are B. All B are C. Some C are D. Conclusions: I. All A are C. II. Some D are A.

Answers

Problem 1: Books inside Pens. Pens and Erasers overlap. Conclusion I: The Books-part of Pens may or may not overlap with the Erasers-part. Not definite. I doesn’t follow. Conclusion II: Some Pens are Erasers is given directly by statement 2. II follows. Answer: Only II follows.

Problem 2: Cats and Dogs are separate. Dogs inside Rats. Conclusion I: Cats are separate from Dogs, but Rats is bigger than Dogs — Cats could overlap with the non-Dog part of Rats. So “No cat is a rat” is not definite. I doesn’t follow. Conclusion II: All Dogs are Rats means the Dogs are inside Rats, so some Rats (namely the Dogs) are definitely Dogs. II follows. Answer: Only II follows.

Problem 3: A inside B. B inside C. So A is inside C (transitive). C and D overlap. Conclusion I: All A are C → A is inside B, B is inside C, so A is inside C. I follows. Conclusion II: Some D are A → D overlaps with C, but the D-C overlap might not include the A part. Not definite. II doesn’t follow. Answer: Only I follows.