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Three-Statement Chain Syllogism

Introduction

Three-Statement Chain Syllogism दो-statement format से आगे बढ़कर एक intermediate term जोड़ता है, जो दो अन्य sets को आपस में जोड़ता है। यह pattern आपकी क्षमता को परखता है कि आप कैसे logical chain relations को पहचानते हैं और कई premises से transitive conclusions निकालते हैं।

यह reasoning tests (banking, SSC, CAT आदि) में आम है, जहां आपको दो extreme terms के बीच final relation infer करना होता है।

Pattern: Three-Statement Chain Syllogism

Pattern

मुख्य अवधारणा: जब दो statements में एक common middle term होता है, तो direction और type सही होने पर first और last terms के बीच नया relation निकाला जा सकता है।

उदाहरण formula: यदि All A are B और All B are C, तो All A are C. यह rule “Some” और “No” statements पर भी लागू होता है, लेकिन केवल तभी जब दिशा और quantifier compatible हों।

Step-by-Step Example

Question

Statements:
1️⃣ All dogs are animals.
2️⃣ All animals are living beings.
3️⃣ All living beings are organisms.

Conclusions:
I. All dogs are living beings.
II. All dogs are organisms.

Options:
A. Only Conclusion I follows
B. Only Conclusion II follows
C. Both I and II follow
D. Neither I nor II follows

Solution

  1. Step 1: Statements को जोड़ें

    All dogs ⊂ Animals; All animals ⊂ Living beings; All living beings ⊂ Organisms.

  2. Step 2: Chain relation बनाएँ

    यह chain - Dogs → Animals → Living beings → Organisms - continuous “All” type relation को दर्शाती है। इससे Dogs का संबंध दोनों से बनता है: Living beings और Organisms।

  3. Step 3: Conclusions evaluate करें

    (I) “All dogs are living beings” - सही। ✅ (II) “All dogs are organisms” - extended transitivity से यह भी सही। ✅

  4. Final Answer:

    Both I and II follow. → Option C

  5. Quick Check:

    All → All → All ⇒ All (Transitive chain fully valid) ✅

Quick Variations

1. All + All ⇒ Strong All relation बनता है।

2. All + Some ⇒ “Some” type relation मिलता है।

3. All + No ⇒ Middle term conflict होने से relation नहीं बनता।

4. Some + Some ⇒ कोई definite conclusion नहीं।

5. Middle term का एक बार subject और एक बार predicate होना आवश्यक है।

Trick to Always Use

  • Middle term खोजें - यही दोनों statements को जोड़ता है।
  • देखें कि दोनों statements एक ही logical direction में जा रही हैं या नहीं।
  • Transitivity rule लागू करें - All → All = All, All → Some = Some, आदि।
  • Direction या quantity mismatch होने पर conclusion follow नहीं करेगा।

Summary

Summary

  • Valid chain तभी बनती है जब middle term linking position (subject/predicate) में सही बैठता हो।
  • Transitive “All” relations सबसे strong universal conclusions देते हैं।
  • “Some” या “No” आने पर chain कमजोर या टूट सकती है।
  • Conclusion पर आने से पहले direction + quantifier दोनों check करें।

याद रखने के लिए उदाहरण:
All A are B; All B are C ⇒ All A are C ✅

Practice

(1/5)
1. Statements: 1️⃣ All cats are mammals. 2️⃣ All mammals are animals. 3️⃣ Some animals are wild. Conclusions: I. Some cats are wild. II. All cats are animals.
easy
A. Only Conclusion I follows
B. Only Conclusion II follows
C. Both I and II follow
D. Neither I nor II follows

Solution

  1. Step 1: Link relations

    All Cats ⊂ Mammals ⊂ Animals.

  2. Step 2: Evaluate Conclusion I

    'Some animals are wild' is an existential about Animals; this does not guarantee that the particular animals which are wild include cats. So Conclusion I does not follow. ❌

  3. Step 3: Evaluate Conclusion II

    All Cats ⊂ Mammals and All Mammals ⊂ Animals ⇒ All Cats ⊂ Animals. So Conclusion II follows. ✅

  4. Final Answer:

    Only Conclusion II follows. → Option B

  5. Quick Check:

    All→All yields universal inclusion; an existential about the superset doesn't force overlap with the subset. ✅
Hint: An existential about the superset doesn't imply the subset is part of that existential.
Common Mistakes: Assuming 'Some animals are X' automatically includes every subset of animals.
2. Statements: 1️⃣ Some birds are parrots. 2️⃣ All parrots are talkative. 3️⃣ All talkative beings are noisy. Conclusions: I. Some birds are noisy. II. All noisy beings are parrots.
easy
A. Both I and II follow
B. Only Conclusion I follows
C. Only Conclusion II follows
D. Neither I nor II follows

Solution

  1. Step 1: Form the forward chain

    Some Birds ↔ Parrots; All Parrots ⊂ Talkative; All Talkative ⊂ Noisy.

  2. Step 2: Evaluate Conclusion I

    Some Birds are Parrots → those Parrots are Talkative → therefore those Parrots are Noisy ⇒ Some Birds are Noisy. ✅

  3. Step 3: Evaluate Conclusion II

    'All noisy beings are parrots' is the reverse/universalization of the forward chain and is not supported. ❌

  4. Final Answer:

    Only Conclusion I follows. → Option B

  5. Quick Check:

    Some→All→All gives a forward 'Some'. Reverse universals do not follow. ✅
Hint: Some + All + All → forward Some at extremes.
Common Mistakes: Reversing forward transitive logic into universals.
3. Statements: 1️⃣ All pens are instruments. 2️⃣ Some instruments are musical. 3️⃣ All musical things are enjoyable. Conclusions: I. Some pens are enjoyable. II. Some enjoyable things are instruments.
easy
A. Only Conclusion II follows
B. Both I and II follow
C. Only Conclusion I follows
D. Neither I nor II follows

Solution

  1. Step 1: Map the chain

    All Pens ⊂ Instruments; Some Instruments ↔ Musical; All Musical ⊂ Enjoyable.

  2. Step 2: Evaluate Conclusion I

    Some Instruments are Musical, but those musical instruments may or may not include Pens. So 'Some pens are enjoyable' is not guaranteed. ❌

  3. Step 3: Evaluate Conclusion II

    Some Instruments are Musical and Musical ⊂ Enjoyable ⇒ those Instruments that are Musical are Enjoyable. Thus there exist Enjoyable things that are Instruments ⇒ Some enjoyable things are instruments. ✅

  4. Final Answer:

    Only Conclusion II follows. → Option A

  5. Quick Check:

    All + Some + All gives a guaranteed 'Some' in the middle → Some enjoyable things are instruments. ✅
Hint: Existential in the middle yields 'Some' about the middle and its supersets/subsets only when overlap includes them.
Common Mistakes: Assuming a 'Some' relationship includes all subsets of the superset.
4. Statements: 1️⃣ All books are papers. 2️⃣ All papers are materials. 3️⃣ Some materials are recyclable. Conclusions: I. Some books are recyclable. II. All recyclable things are papers.
medium
A. Only Conclusion I follows
B. Only Conclusion II follows
C. Both I and II follow
D. Neither I nor II follows

Solution

  1. Step 1: Form chain

    All Books ⊂ Papers ⊂ Materials.

  2. Step 2: Evaluate Conclusion I

    'Some materials are recyclable' does not guarantee that the recyclable materials include books. The recyclable portion might be disjoint from books, so Some books are recyclable is not certain. ❌

  3. Step 3: Evaluate Conclusion II

    'All recyclable things are papers' is not supported; recyclable things are only a subset of materials and may or may not all be papers. ❌

  4. Final Answer:

    Neither I nor II follows. → Option D

  5. Quick Check:

    Subset chain + existential at superset does not ensure overlap with the subset. ✅
Hint: An existential about the superset can't be assumed to include a particular subset unless stated.
Common Mistakes: Assuming some property of the superset applies to every subset.
5. Statements: 1️⃣ Some engineers are designers. 2️⃣ All designers are creative. 3️⃣ Some creative people are artists. Conclusions: I. Some engineers are artists. II. Some artists are designers.
medium
A. Both I and II follow
B. Neither I nor II follows
C. Only Conclusion I follows
D. Only Conclusion II follows

Solution

  1. Step 1: Understand chain

    Some Engineers ↔ Designers; All Designers ⊂ Creative; Some Creative ↔ Artists.

  2. Step 2: Evaluate Conclusion I

    We have Some Engineers that are Designers and All Designers are Creative; Some Creative are Artists - but there's no guarantee that the particular Creative persons who are Artists overlap with the particular Designers who are Engineers. So 'Some engineers are artists' is not certain. ❌

  3. Step 3: Evaluate Conclusion II

    'Some artists are designers' is also not guaranteed: Some Creative are Artists and All Designers are Creative, but that does not force Designers to overlap the Artists. ❌

  4. Final Answer:

    Neither I nor II follows. → Option B

  5. Quick Check:

    Two separate existential facts about the same middle set do not guarantee overlap between the corresponding particular parts. ✅
Hint: Avoid chaining multiple 'Some' facts expecting guaranteed overlap unless one existential explicitly includes the other.
Common Mistakes: Assuming existence in a superset implies overlap across different existential parts.

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