Syllogism — Study Notes for RRB NTPC

Overview

Syllogism tests your ability to draw valid logical conclusions from two or more statements without relying on general knowledge or assumptions. In RRB NTPC, you will encounter 2–3 statement syllogisms using quantifiers like "All," "Some," and "No." Each problem provides premises (statements) and asks whether given conclusions logically follow.

This topic appears regularly in RRB NTPC with 2–4 questions per exam. Mastering syllogism requires understanding Venn diagram representations and the rules of logical deduction. Many students lose marks by applying real-world knowledge instead of pure logic or by misinterpreting the meaning of "some." The key skill is translating statements into diagrams and mechanically checking whether conclusions are necessarily true, possibly true, or definitely false.

Success in syllogism problems comes from practice and method, not intuition. Use Venn diagrams consistently, learn the standard valid and invalid patterns, and never assume information beyond what the statements explicitly provide.

Key Concepts

• **Universal Affirmative (All A are B)**: Every member of set A is contained within set B. Venn diagram shows circle A completely inside circle B.

• **Universal Negative (No A are B)**: Sets A and B have no common members. Venn diagram shows two separate, non-overlapping circles.

• **Particular Affirmative (Some A are B)**: At least one member of A is also a member of B. Venn diagram shows overlapping circles with intersection marked; "some" means "at least one," not "only a few."

• **Particular Negative (Some A are not B)**: At least one member of A is outside B. Part of circle A extends beyond circle B.

• **Complementary pairs**: "All A are B" and "Some A are not B" cannot both be true. "No A are B" and "Some A are B" cannot both be true. These help in either-or conclusion questions.

• **Valid conclusion rule**: A conclusion is valid only if it is true in every possible Venn diagram representation of the premises. If you can draw even one diagram where the premises hold but the conclusion fails, the conclusion is invalid.

• **"Some" is bidirectional**: "Some A are B" automatically means "Some B are A." Both statements are logically equivalent and interchangeable.

Formulas / Key Facts

1. **All A are B + All B are C → All A are C** (transitive chain for universal affirmatives) 2. **All A are B + No B are C → No A are C** (universal affirmative + universal negative) 3. **Some A are B + All B are C → Some A are C** (particular carried through universal) 4. **No A are B + All B are C → No A are C** (negative carried through universal) 5. **Two particular statements (Some/Some not) yield no definite conclusion** — never combine two "some" statements to reach a valid conclusion 6. **Two negative premises (No/Some not) yield no definite conclusion** — you cannot derive a positive relationship from two negatives 7. **If no direct conclusion follows, check for complementary pair** — you may have "Either conclusion I or conclusion II follows" 8. **Conversion rules**: "All A are B" does NOT convert to "All B are A"; "No A are B" converts to "No B are A"; "Some A are B" converts to "Some B are A"

Worked Examples

**Example 1: Standard two-statement syllogism**

*Statements:* All mangoes are fruits. All fruits are sweet.

*Conclusions:* I. All mangoes are sweet. II. Some sweet things are fruits.

*Solution:* Draw Venn diagram: mangoes ⊂ fruits ⊂ sweet (three concentric circles). Check conclusion I: Since all mangoes are inside fruits and all fruits are inside sweet, all mangoes must be sweet. **Conclusion I follows.** Check conclusion II: All fruits are sweet means some sweet things are definitely fruits. **Conclusion II follows.** Answer: Both conclusions follow.

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**Example 2: Negative statement**

*Statements:* No pen is pencil. All pencils are erasers.

*Conclusions:* I. No pen is eraser. II. Some erasers are not pens.

*Solution:* Draw diagram: Pen and pencil are separate circles. Pencil circle is completely inside eraser circle. Check conclusion I: Erasers contain all pencils, but erasers might also overlap with pens. We cannot definitively say no pen is eraser. **Conclusion I does not follow.** Check conclusion II: Since all pencils are erasers and no pen is pencil, at least those pencils (which are erasers) are not pens. **Conclusion II follows.** Answer: Only conclusion II follows.

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**Example 3: Complementary pair (either-or case)**

*Statements:* All cats are animals. All animals are dogs.

*Conclusions:* I. All cats are dogs. II. Some cats are not dogs.

*Solution:* From statements: cats ⊂ animals ⊂ dogs, so all cats are dogs. Check conclusion I: **Follows directly.** Check conclusion II: This contradicts conclusion I. **Does not follow.** Since I and II form a complementary pair (All vs Some not) and statement chain proves "All cats are dogs," only conclusion I follows. Answer: Only conclusion I follows.

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**Example 4: Three-statement syllogism**

*Statements:* Some books are novels. All novels are stories. No story is a poem.

*Conclusions:* I. Some books are stories. II. No novel is a poem.

*Solution:* Some books overlap with novels; all novels inside stories; stories and poems separate. Check conclusion I: Since some books are novels and all novels are stories, some books must be stories. **Conclusion I follows.** Check conclusion II: All novels are stories, and no story is poem, so no novel is poem. **Conclusion II follows.** Answer: Both conclusions follow.

Common Mistakes

• **Applying real-world knowledge → Ignore what you know about the actual world**: If the statement says "All tables are chairs," accept it as true for the problem even though it contradicts reality. Base conclusions only on given statements.

• **Misunderstanding "some" as "only some" or "a few" → "Some" means "at least one"**: "Some A are B" does not exclude the possibility that all A are B. It simply confirms at least one overlap.

• **Assuming "All A are B" means "All B are A" → Converse is not automatic**: The statement "All mangoes are fruits" does not imply "All fruits are mangoes." Never reverse a universal affirmative unless explicitly stated.

• **Combining two "some" statements to reach a conclusion → Two particulars yield nothing definite**: "Some A are B" and "Some B are C" do not allow you to conclude anything about A and C. You need at least one universal statement for a valid chain.

• **Forgetting to check complementary pairs in either-or questions → Check if conclusions are exact opposites**: When no single conclusion follows, see if one conclusion is "All A are B" and the other is "Some A are not B" (or "No A are B" vs "Some A are B"). If so, one of them must be true — answer "Either I or II follows."

Quick Reference

• Use Venn diagrams for every problem — visual method eliminates errors.
• "All A are B" = A circle inside B circle.
• "No A are B" = A and B circles completely separate.
• "Some A are B" = A and B circles overlap; at least one element in intersection.
• Two "some" statements or two "no" statements → no valid conclusion.
• Always check for complementary pairs if individual conclusions fail.