न्याय-वाक्य (Syllogism)

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Syllogism (न्याय-वाक्य)

What You Will Learn

By the end of this chapter, you will be able to:

• Understand what a syllogism question is and how to solve it.
• Identify the four types of statements: All, No, Some, Some Not.
• Draw Venn diagrams for statements and test conclusions against them.
• Handle special cases: possibility conclusions, complementary pairs (either–or), and negative conclusions.
• Avoid the traps examiners set in SSC, Railway, and Bank exams.

1. What Is a Syllogism?

A syllogism is a logical argument made of:

1. Statements (Premises) — two or more sentences that you must accept as true, even if they contradict real-world facts.
2. Conclusions — sentences that you must test: do they definitely follow from the statements?

Golden rule: Forget what you know about the real world. If the statement says "All cats are dogs," then inside that question, all cats really are dogs.

Example:

• Statements: All pens are books. All books are chairs.
• Conclusion: All pens are chairs. ✅ (definitely follows)

2. The Four Types of Statements

Key facts to memorise:

• "Some" means at least one — it can also mean all. So if "All A are B" is true, "Some A are B" is also true.
• "No A is B" and "No B is A" mean the same thing (both circles are separate).
• "All A are B" does not mean "All B are A." (All apples are fruits, but not all fruits are apples.)
• "Some A are B" does not automatically mean "Some A are not B."

3. Solving with Venn Diagrams — Step by Step

Step 1: Draw the diagram for each statement. Draw the minimum arrangement first.

Step 2: For each conclusion, ask: *Is this true in every possible diagram I could draw from these statements?*

• If yes in all cases → the conclusion follows.
• If it fails in even one possible diagram → the conclusion does not follow.

Worked Example 1:

• Statements: All dogs are animals. Some animals are cats.
• Conclusions:

1. Some dogs are cats. ❌ (The "some animals" that are cats may lie outside the dogs circle — not definite.)
2. All dogs are animals. ✅ (Directly given.)

Worked Example 2 (negative statement):

• Statements: No pen is a pencil. All pencils are erasers.
• Conclusions:

1. No pen is an eraser. ❌ (The eraser circle may or may not touch the pen circle — not definite.)
2. Some erasers are not pens. ✅ (The part of erasers inside pencils can never be pens, since no pen is a pencil.)

4. Special Cases

4.1 Complementary Pairs (Either–Or)

Two conclusions form an either–or pair when:

• Both are doubtful (neither definitely follows alone), and
• They have the same subject and predicate, and
• One is positive and the other negative (e.g., "Some A are B" + "No A is B", or "Some A are B" + "Some A are not B").

Then the answer is "either conclusion I or II follows."

4.2 Possibility Conclusions

A conclusion worded as a possibility ("Some A being B is a possibility") follows if the diagram does not forbid it — even if it is not definite. But if a definite conclusion already follows, the possibility version is usually not needed, and if the statement forbids it (e.g., "No A is B"), the possibility is false.

4.3 Definite vs. Possible — the examiner's favourite trap

• "Some A are B" (definite) needs proof in every diagram.
• "Possibly some A are B" needs only one allowed diagram.

5. Common Traps in Exams

1. Real-world knowledge trap: Statements may be absurd ("All tables are dogs"). Accept them anyway.
2. Reversal trap: "All A are B" does not give "All B are A."
3. Some ≠ Some not: From "Some A are B" alone, you cannot conclude "Some A are not B."
4. Middle-term jump: "All A are B" + "All C are B" gives no definite relation between A and C.
5. Negative statement trap: With "No A is B," any positive definite conclusion between A and B is false, but negative conclusions like "Some B are not A" may follow.

6. PYQ Pattern Analysis (परीक्षा पैटर्न)

*(Pattern analysis in words — no invented counts.)*

• SSC (CGL, CHSL, MTS, GD) and RRB (NTPC, Group D): Usually 2 statements + 2 conclusions, all four statement types mixed. Questions are direct; Venn diagrams solve them in under a minute.
• Bank exams (IBPS, SBI): 3–4 statements, "only conclusion I / only II / both / neither / either–or" options, plus possibility conclusions and reverse-syllogism style items.
• Frequently repeated sub-topics: complementary pairs (either–or), "Some are not" conclusions, and questions where a real-world-true conclusion does not logically follow.
• Examiner's traps: near-correct conclusions that are only *possible* but not *definite*; conclusions that reverse "All A are B"; negative-worded options ("which does NOT follow").

Time tip: Aim for 30–45 seconds per question. Draw small, quick diagrams — neatness does not earn marks.

7. Mind Map

• Syllogism

  • Statements
    (always taken as true)

    • All A are B
    • No A is B
    • Some A are B
    • Some A are not B
  • Conclusions
    (must definitely follow)

    • Definite:
      true in every diagram
    • Possibility:
      true in at least one diagram
    • Either-Or:
      complementary pair
  • Method: Venn Diagrams
  • Traps:
    reversal, real-world facts,
    some vs some-not

8. Flashcards

Card 1
Front: What is a syllogism?
Back: A logical argument with statements (taken as true) and conclusions that must be tested for whether they definitely follow.

Card 2
Front: What does "Some A are B" mean in logic?
Back: At least one A is B. It can include the case where all A are B.

Card 3
Front: Does "All A are B" imply "All B are A"?
Back: No. Reversal is not valid (all apples are fruits, but not all fruits are apples).

Card 4
Front: What is the diagram for "No A is B"?
Back: Two completely separate circles — A and B share nothing.

Card 5
Front: When does a definite conclusion follow?
Back: When it is true in every possible Venn diagram drawn from the statements.

Card 6
Front: When does a possibility conclusion follow?
Back: When at least one valid diagram allows it and no statement forbids it.

Card 7
Front: What is a complementary (either–or) pair?
Back: Two doubtful conclusions with the same subject and predicate, one positive and one negative — then "either I or II follows."

Card 8
Front: From "Some A are B," can we conclude "Some A are not B"?
Back: No, not by itself. "Some" may include all.

Card 9
Front: Statements: All A are B; All B are C. Conclusion: All A are C?
Back: Yes — it definitely follows (chain of inclusion).

Card 10
Front: Statements: All A are B; All C are B. Any definite relation between A and C?
Back: No. Both being inside B does not connect A and C.

Card 11
Front: Should you use real-world knowledge in syllogism?
Back: Never. Accept the statements as true even if they sound absurd.

Card 12
Front: "No A is B" — is "No B is A" also true?
Back: Yes. Separation is mutual.

Card 13
Front: Statements: No pen is pencil; All pencils are erasers. What follows definitely?
Back: Some erasers are not pens. (No pen is eraser does NOT follow.)

Card 14
Front: Which exams ask syllogism with possibility conclusions?
Back: Bank exams (IBPS, SBI) frequently; SSC/Railway mostly ask definite conclusions.

Card 15
Front: Ideal time per syllogism question?
Back: About 30–45 seconds using quick Venn diagrams.

*Source: Standard reasoning concepts (NCERT-level logic; pattern based on SSC/RRB/Bank exam trends).*