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Possibility-Based Syllogism

Introduction

Possibility-Based Syllogisms என்பது, கொடுக்கப்பட்ட premises-க்கு முரணாக இல்லாமல் ஒரு stated relation உண்மையாக இருக்க முடியுமா என்பதை சோதிக்கும் pattern ஆகும். எது கண்டிப்பாக (must) வரும் என்று கேட்பதற்குப் பதிலாக, எது சாத்தியம் (possible) என்று கேட்பதே இதன் சிறப்பு - competitive reasoning-ல் அடிக்கடி வரும் ஒரு subtle twist.

Possibility checks-ஐ நன்றாக கற்றுக்கொள்வது முக்கியம். ஏனெனில், பல questions-ல் incomplete information இருக்கும்; எந்த conclusions logically possible ஆக இருக்கின்றன (certainty இல்லாவிட்டாலும்) என்பதை அடையாளம் காண்பதே cautious solvers மற்றும் excellent solvers இடையிலான வித்தியாசம்.

Pattern: Possibility-Based Syllogism

Pattern

The key idea: A conclusion is a possibility if it does not contradict any given premise; it is impossible if it directly contradicts a premise.

பயன்படுத்த வேண்டிய விதிகள்:

  • ஒரு conclusion, explicit universal (“All” / “No”) statement-ஐ மீறினால், அது impossible.
  • Premises-ல் எதுவும் ஒரு conclusion-ஐ contradict செய்யாவிட்டால், (premises opposite-ஐ force செய்யாவிட்டால்) அதை possible என்று கருத வேண்டும்.
  • Possibility questions-ல் பொதுவாக “possible,” “can,” அல்லது conclusion may be true என்ற வார்த்தைகள் வரும்.
  • follows (must be true) மற்றும் is possible (may be true) ஆகிய இரண்டிற்கும் உள்ள வேறுபாட்டை கவனமாக பிரிக்க வேண்டும்.

Step-by-Step Example

Question

Statements:
1️⃣ All painters are artists.
2️⃣ Some artists are sculptors.

கீழ்கண்டவற்றில் எது possible?
Options:
A. All painters are sculptors.
B. Some painters are sculptors.
C. No painter is a sculptor.
D. All sculptors are painters.
E. All of the above

Solution

  1. Step 1: Restate premises

    All Painters ⊂ Artists. Some Artists ↔ Sculptors (Artists மற்றும் Sculptors இடையே partial overlap உள்ளது).

  2. Step 2: Check Option A

    “All painters are sculptors” ⇒ Painters ⊂ Sculptors. இதைத் தடை செய்யும் universal premise எதுவும் இல்லை; ஆகவே இது possible. ✅

  3. Step 3: Check Option B

    “Some painters are sculptors” என்பதும் possible (Painters, overlap பகுதியில் இருந்தால் இது நடக்கலாம்). ✅

  4. Step 4: Check Option C

    “No painter is a sculptor” என்பதும் premises-க்கு முரணாக இல்லை: painters, sculptors-க்கு ஒட்டாத artist பகுதியில் இருக்கலாம். ஆகவே இது possible. ✅

  5. Step 5: Check Option D

    “All sculptors are painters” (Sculptors ⊂ Painters) என்பது guarantee இல்லை; ஆனால் கொடுக்கப்பட்ட statements இதை நேரடியாக contradict செய்யவில்லை (நமக்கு தெரியும் என்பது Some artists are sculptors மட்டும்). ஆகவே இது கூட possible. ✅

  6. Final Answer:

    Options A, B, C, மற்றும் D அனைத்தும் possible. → Option E

  7. Quick Check:

    எந்த option-மும் universal premise-ஐ contradict செய்யவில்லை (எ.கா., “No artist is a sculptor” அல்லது “All sculptors are not artists” போன்ற statement இல்லை). ஆகவே நான்கு possibilities-யும் ஏதோ ஒரு scenario-வில் logically hold ஆகலாம். ✅

Quick Variations

1. Possible vs Impossible: “follows” என்பதற்கு பதிலாக “possible” வந்தால் answer pattern மாறும்.

2. Mix of All/No/Some: explicit “No A is B” இருந்தால், எந்த “A can be B” என்பதும் உடனே impossible.

3. Existential constraints: premises-ல் “Some A are B” இருந்தால், “No A is B” impossible ஆகிவிடும்.

4. Multiple premises: possibility என்று அறிவிப்பதற்கு முன், ஒவ்வொரு premise-யையும் direct contradiction-க்கு சோதிக்கவும்.

Trick to Always Use

  • Step 1 → முதலில் universal negatives (“No …”) இருக்கிறதா என்று பாருங்கள் - அவை impossibilities உருவாக்கும்.
  • Step 2 → எந்த premise-மும் candidate conclusion-ஐ நேரடியாக contradict செய்யாவிட்டால், அதை possible என்று எடுத்துக்கொள்ளுங்கள்.
  • Step 3 → நினைவில் வையுங்கள்: “possible” ≠ “follows”; explicitly rule out செய்யப்படாவிட்டால் possible.

Summary

Summary

  • முதலில் universal statements-ஐ அடையாளம் காணுங்கள் - அவை hard impossibilities உருவாக்கும்.
  • ஒரு conclusion எந்த premise-க்கும் முரணாக இல்லையெனில், அது logically possible.
  • “Some” premises existence-ஐ மட்டும் காட்டும்; contradiction இல்லாவிட்டால் alternative possibilities-ஐ eliminate செய்யாது.
  • Candidate conclusion-க்கு எதிராக ஒவ்வொரு premise-யையும் direct contradiction-க்கு மீண்டும் சரிபார்க்கவும்.

Example to remember:
All A are B; Some B are C ⇒ All A are C, அல்லது Some A are C, அல்லது No A are C possible - எந்த premise-மும் தடை செய்யாவிட்டால். ✅

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; All Mammals ⊂ Animals ⇒ All Cats ⊂ Animals.

  2. Step 2: Evaluate Conclusion I

    ‘Some animals are wild’ is an existential about Animals (the superset). That existential does not guarantee that the particular wild animals include Cats (the subset). Therefore ‘Some cats are wild’ is only possible, not certain. ❌

  3. Step 3: Evaluate Conclusion II

    From the universals All Cats ⊂ Mammals and All Mammals ⊂ Animals we can deduce All Cats ⊂ Animals. Hence ‘All cats are animals’ is definitely true. ✅

  4. Final Answer:

    Only Conclusion II follows. → Option B

  5. Quick Check:

    All → All gives a universal inclusion; an existential about the superset does not force overlap with a particular 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 All roses are flowers. 2️⃣ No flower is a tree. Which of the following is impossible?
easy
A. Some roses are trees
B. No rose is a tree
C. All roses are flowers
D. Some trees are not roses

Solution

  1. Step 1: Restate premises

    All Roses ⊂ Flowers; Flowers ∩ Trees = ∅ (flowers and trees are disjoint).

  2. Step 2: Derive consequence

    Since Roses ⊂ Flowers and Flowers ∩ Trees = ∅, it follows that Roses ∩ Trees = ∅ (roses cannot overlap with trees).

  3. Step 3: Determine impossibility

    ‘Some roses are trees’ asserts Roses ∩ Trees ≠ ∅, which directly contradicts Roses ∩ Trees = ∅. Therefore this statement is impossible.

  4. Final Answer:

    Some roses are trees. → Option A

  5. Quick Check:

    Subset of a set disjoint with another set ⇒ the subset is also disjoint with that set → 'Some roses are trees' impossible. ✅
Hint: If A ⊂ B and B ∩ C = ∅ then A ∩ C = ∅ (so any 'Some A are C' is impossible).
Common Mistakes: Treating equivalent phrasings as distinct; ensure you map subsets first.
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 pens are enjoyable’ would require that the particular instruments which are musical include pens. The premises only say some instruments are musical - they do not guarantee that those musical instruments include pens. So Conclusion I is not certain (only possible). ❌

  3. Step 3: Evaluate Conclusion II

    Some Instruments are Musical and Musical ⊂ Enjoyable ⇒ the musical instruments are enjoyable. Therefore there exist Enjoyable things that are Instruments - i.e., Some enjoyable things are instruments. This conclusion follows. ✅

  4. Final Answer:

    Only Conclusion II follows. → Option A

  5. Quick Check:

    All + Some + All produces an existential about the middle that transfers to the superset; the subset (pens) only becomes existential if explicitly overlapping the 'Some' portion. ✅
Hint: A 'Some' in the middle yields a guaranteed 'Some' for related supersets/subsets only when overlap is specified.
Common Mistakes: Assuming the middle 'Some' automatically includes all subsets of the superset.
4. Statements: 1️⃣ No poet is a banker. 2️⃣ Some bankers are writers. Which of the following is impossible?
medium
A. Some poets are writers
B. All writers are poets
C. Some writers are bankers
D. No writer is a banker

Solution

  1. Step 1: Restate premises

    No Poet ⟂ Banker; Some Bankers ↔ Writers (existential overlap between Bankers and Writers).

  2. Step 2: Test each option

    A (Some poets are writers) is possible because poets could be writers and remain non-bankers; B (All writers are poets) is not contradicted by premises (writers could all be poets who are not bankers) - it's extreme but possible; C (Some writers are bankers) follows from premise 2 and is therefore possible/true. D (No writer is a banker) directly contradicts premise 2 (Some bankers are writers implies some writers are bankers), so D is impossible.

  3. Final Answer:

    No writer is a banker. → Option D

  4. Quick Check:

    'Some bankers are writers' rules out any universal claim that 'No writer is a banker'. ✅
Hint: An existential ('Some ...') immediately invalidates the opposite universal ('No ...').
Common Mistakes: Confusing possibility with necessity: extreme universal claims may still be logically possible unless directly contradicted by an existential.
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

    ‘Some engineers are artists’ would require the particular designers who are engineers to overlap with the particular creative people who are artists. The premises do not guarantee this overlap - it is only possible, not certain. ❌

  3. Step 3: Evaluate Conclusion II

    ‘Some artists are designers’ would require the particular creative people who are artists to overlap with the particular designers. The premises state Some Creative are Artists and All Designers ⊂ Creative, but that does not force overlap between Artists and Designers. Therefore Conclusion II is not guaranteed either. ❌

  4. Final Answer:

    Neither I nor II follows. → Option B

  5. Quick Check:

    Two separate existential facts about different parts of the same superset do not ensure those particular parts overlap. ✅
Hint: Avoid chaining multiple 'Some' facts into certainty unless an explicit overlap is given.
Common Mistakes: Assuming existence in the superset implies overlap across different existential parts.

Mock Test

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