Syllogism

Syllogism — Study Notes for MPESB Group 1/2/3

Overview

Syllogism is a topic in the Reasoning and Aptitude section of MPESB exams. It tests your ability to draw valid conclusions from given statements, regardless of real-world truth. The logic is formal—what matters is the structure of the argument, not whether "All cats are dogs" makes sense in reality.

Syllogism questions typically present 2–3 statements followed by 2–4 conclusions. You must identify which conclusions logically follow. The Venn diagram method is the most reliable approach for solving these questions quickly and accurately.

The key skill here is systematic representation. Students who try to solve syllogism "by intuition" often fall into logical traps.

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Key Concepts

• Statement vs. Conclusion: Statements are given premises (accept them as true). Conclusions are claims you must test against the statements—never add outside knowledge.
• Universal Statements: "All A are B" and "No A is B" apply to the entire group. Universal Affirmative (All) and Universal Negative (No) are definite claims.
• Particular Statements: "Some A are B" and "Some A are not B" apply to at least one member. Particular Affirmative (Some) and Particular Negative (Some not) are partial claims.
• "Some" means "at least one": When a statement says "Some A are B," it could mean one, many, or even all A are B. This is critical for understanding possibilities.
• Definite vs. Possible Conclusions: A definite conclusion must be true in ALL valid Venn diagram representations. A possible conclusion is true in AT LEAST ONE valid representation.
• Complementary Pair Rule: "Some A are B" and "Some A are not B" form a complementary pair—at least one of them is always true when dealing with two overlapping categories.
• Conversion Rules: "All A are B" converts to "Some B are A." "Some A are B" converts to "Some B are A." "No A is B" converts to "No B is A." "Some A are not B" does NOT convert.

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Key Facts and Rules

The Four Standard Propositions:

Venn Diagram Representation:

• "All A are B" → Circle A completely inside Circle B
• "No A is B" → Circles A and B do not touch at all
• "Some A are B" → Circles A and B overlap partially
• "Some A are not B" → Part of Circle A lies outside Circle B

Valid Inference Rules:

• All A are B + All B are C → All A are C (valid)
• All A are B + Some B are C → Some A are C (NOT valid—no definite conclusion)
• No A is B + All B are C → Some C are not A (valid)
• Some A are B + Some B are C → No definite conclusion possible

Either-Or Rule: When neither conclusion follows individually, check if they form a complementary pair. If yes, answer is "Either I or II follows."

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Worked Examples

Example 1: Basic Two-Statement Problem

Statements:

1. All pens are pencils.
2. All pencils are erasers.

Conclusions: I. All pens are erasers. II. Some erasers are pens.

Solution:

• Draw Circle "Pens" inside Circle "Pencils"
• Draw Circle "Pencils" inside Circle "Erasers"
• Result: Pens is completely inside Erasers
• Conclusion I: All pens are erasers → Follows ✓
• Conclusion II: Some erasers are pens → Follows ✓ (conversion of I)

Answer: Both I and II follow

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Example 2: Universal Negative

Statements:

1. No book is a pen.
2. All pens are chairs.

Conclusions: I. No book is a chair. II. Some chairs are not books.

Solution:

• Draw Circle "Book" and Circle "Pen" with no overlap
• Draw Circle "Pen" completely inside Circle "Chair"
• Books and Chairs may or may not overlap (draw multiple possibilities)
• Conclusion I: No book is a chair → Does not definitely follow (they might overlap)
• Conclusion II: Some chairs are not books → Follows ✓ (the pen portion of chairs is definitely not books)

Answer: Only II follows

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Example 3: Possibility-Based Question

Statements:

1. Some cats are dogs.
2. All dogs are horses.

Conclusions: I. All cats being horses is a possibility. II. Some cats are horses.

Solution:

• Draw overlapping circles for Cats and Dogs
• Draw Dogs completely inside Horses
• The overlapping portion of Cats-Dogs is inside Horses
• Conclusion I: Can all cats be inside horses? Yes, it's possible. → Follows ✓
• Conclusion II: The cat-dog overlap is definitely inside horses. → Follows ✓

Answer: Both I and II follow

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Common Mistakes

Mistake 1: Applying real-world logic Wrong thinking: "All cats are dogs" is false, so I'll assume it's wrong. Correct fix: Accept all statements as absolutely true within the problem. Real-world facts are irrelevant.

Mistake 2: Confusing "Some" with "Only Some" Wrong thinking: "Some A are B" means only a few A are B, not all. Correct fix: "Some" means "at least one"—it includes the possibility that all A are B. Draw multiple Venn diagrams if needed.

Mistake 3: Drawing only one Venn diagram Wrong thinking: Drawing one diagram and checking conclusions against it. Correct fix: For "Some" statements, draw all possible valid representations. A conclusion follows only if true in ALL diagrams.

Mistake 4: Missing the Either-Or situation Wrong thinking: If neither conclusion follows individually, both are wrong. Correct fix: Check if conclusions form a complementary pair (Some A are B / Some A are not B). If yes, answer is "Either I or II follows."

Mistake 5: Incorrectly converting "Some A are not B" Wrong thinking: "Some A are not B" means "Some B are not A." Correct fix: "Some A are not B" cannot be converted. This is the only proposition type that has no valid conversion.

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Quick Reference

• All = Circle inside Circle | No = Circles apart | Some = Circles overlap
• "Some A are B" always converts to "Some B are A"—both are identical in Venn terms.
• Universal statements (All/No) give definite conclusions; Particular statements (Some) often give no definite conclusion when combined.
• When stuck between two negative conclusions, check for the complementary pair (Either-Or).
• For "possibility" questions: if you can draw even ONE valid diagram where the conclusion holds, it's possible.
• "Some A are not B" is the troublemaker—it doesn't convert and rarely gives clean conclusions.