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Deduction with Multiple Premises

Introduction

logical reasoning-ல், பல கேள்விகளில் இரண்டு அல்லது அதற்கு மேற்பட்ட premises-ஐ இணைத்து சரியான conclusion-க்கு வர வேண்டும். இந்த கேள்விகள், statements-ஐ படிப்படியாக இணைத்து - பல premises-களுக்குள் ஒரு logical chain உருவாக்கும் உங்கள் திறனை சோதிக்கின்றன.

இந்த pattern முக்கியமானது; ஏனெனில் இது transitive reasoning-ஐ வளர்க்கிறது மற்றும் சிக்கலான syllogisms மற்றும் inference-based sets-ஐ தீர்க்க உதவுகிறது.

Pattern: Deduction with Multiple Premises

Pattern

முக்கிய கருத்து: கொடுக்கப்பட்ட premises (facts)-ஐ logically இணைப்பது - ஒரு statement மற்றொன்றை நோக்கி வழிநடத்தும் போது - கட்டாயமாக follow ஆகும் புதிய conclusion-ஐ பெறுவது.

உதாரணமாக, A → B மற்றும் B → C என்றால், நாம் சரியாக A → C என முடிவு செய்யலாம். இதையே Transitive Deduction என்று அழைக்கப்படுகிறது.

Step-by-Step Example

Question

Premises:
1️⃣ எல்லா fruits-உம் food ஆகும்.
2️⃣ எல்லா apples-உம் fruits ஆகும்.
3️⃣ எல்லா food items-உம் edible ஆகும்.

எந்த conclusion கண்டிப்பாக follow ஆகிறது?
(A) எல்லா fruits-உம் apples ஆகும்.
(B) எல்லா apples-உம் edible ஆகும்.
(C) சில apples edible அல்ல.
(D) எல்லா edible items-உம் apples ஆகும்.

Solution

  1. Step 1: முதல் இரண்டு premises-ஐ இணைக்கவும்

    All apples ⊂ fruits; all fruits ⊂ food ⇒ All apples ⊂ food.

  2. Step 2: மூன்றாவது premise-ஐ சேர்க்கவும்

    All food ⊂ edible ⇒ All apples ⊂ edible.

  3. Step 3: மற்ற options-ஐ சரிபார்க்கவும்

    ‘All fruits are apples’ ❌ (reverse), ‘Some apples not edible’ ❌ (given-க்கு முரண்படும்), ‘All edible are apples’ ❌ (அதிகமாக பொதுவாக்கப்பட்டது).

  4. Final Answer:

    All apples are edible → Option B

  5. Quick Check:

    Chain - Apples → Fruits → Food → Edible ✅

Quick Variations

1. 3 அல்லது அதற்கு மேற்பட்ட premises-களுடன் transitive deduction.

2. negative statements உள்ள mixed premises (உதா: “No A is B, All B are C”).

3. indirect அல்லது reversed logical chains.

4. பல links-களில் வரும் “if-then” வடிவ conditional deductions.

Trick to Always Use

  • Step 1: premises-ஐ சரியான வரிசையில் அமைக்கவும் (A → B → C).
  • Step 2: பொதுவான terms-ஐ இணைத்து logical chain உருவாக்கவும்.
  • Step 3: direction-ஐ சரிபார்க்கவும் - forward logical flow-ல் மட்டுமே valid.
  • Step 4: reverse அல்லது தொடர்பில்லாத statements-ஐ உடனே நிராகரிக்கவும்.

Summary

Summary

  • பல premises-களை shared terms மூலம் மட்டுமே இணைக்க முடியும்.
  • Transitive logic, எல்லா links-உம் ஒரே திசையில் இணைந்தால் மட்டுமே வேலை செய்யும்.
  • negative statements direct chain-ஐ உடைக்கும் - கவனமாக கையாளவும்.
  • எப்போதும் முதல் subject-இலிருந்து இறுதி object வரை வாசிக்கவும்.

நினைவில் வைக்க வேண்டிய example:
If All A → B, All B → C, and All C → D ⇒ All A → D ✅

Practice

(1/5)
1. Premises:<br>1️⃣ All poets are writers.<br>2️⃣ All writers are readers.<br>3️⃣ Some readers are thinkers.<br><br>Which of the following conclusions definitely follows?<br>(A) All poets are readers.<br>(B) Some poets are thinkers.<br>(C) All thinkers are readers.<br>(D) Some writers are not poets.
easy
A. All poets are readers.
B. Some poets are thinkers.
C. All thinkers are readers.
D. Some writers are not poets.

Solution

  1. Step 1: Link Premises

    All poets ⊂ writers, and all writers ⊂ readers ⇒ All poets ⊂ readers.

  2. Step 2: Evaluate Other Options

    ‘Some poets are thinkers’ - not directly stated → Uncertain. ‘All thinkers are readers’ - not given. ‘Some writers are not poets’ - not stated.

  3. Final Answer:

    All poets are readers → Option A

  4. Quick Check:

    Poets → Writers → Readers ✅
Hint: Combine consecutive ‘All’ statements to form a clear chain.
Common Mistakes: Assuming extra links like ‘poets are thinkers’.
2. Premises:<br>1️⃣ All roses are flowers.<br>2️⃣ Some flowers are red.<br>3️⃣ All red things are attractive.<br><br>Which of the following conclusions definitely follows?<br>(A) All flowers are attractive.<br>(B) Some flowers are attractive.<br>(C) Some red things are roses.<br>(D) All red things are flowers.
easy
A. All flowers are attractive.
B. Some flowers are attractive.
C. Some red things are roses.
D. All red things are flowers.

Solution

  1. Step 1: Link Premises

    Some flowers are red; all red things are attractive ⇒ Some flowers are attractive.

  2. Step 2: Eliminate Others

    ‘All flowers are attractive’ ❌ not stated. ‘Some red things are roses’ ❌ not given. ‘All red things are flowers’ ❌ reverse.

  3. Final Answer:

    Some flowers are attractive → Option B

  4. Quick Check:

    Flowers → Red → Attractive ✅
Hint: ‘Some + All’ combination yields ‘Some’ relation at the end.
Common Mistakes: Extending ‘some’ to mean ‘all’.
3. Premises:<br>1️⃣ All students are learners.<br>2️⃣ Some learners are teachers.<br>3️⃣ All teachers are professionals.<br><br>Which of the following conclusions definitely follows?<br>(A) Some students are professionals.<br>(B) Some professionals are learners.<br>(C) All learners are professionals.<br>(D) All professionals are students.
medium
A. Some students are professionals.
B. Some professionals are learners.
C. All learners are professionals.
D. All professionals are students.

Solution

  1. Step 1: Link Premises

    Some learners ⊂ teachers; all teachers ⊂ professionals ⇒ Some learners ⊂ professionals ⇒ Some professionals ⊂ learners.

  2. Step 2: Evaluate

    ‘Some professionals are learners’ follows from reverse inclusion → ✅ True.

  3. Final Answer:

    Some professionals are learners → Option B

  4. Quick Check:

    Learners → Teachers → Professionals ✅
Hint: Reverse the ‘Some-All’ chain to get valid ‘Some’ conclusions.
Common Mistakes: Assuming ‘All learners are professionals’.
4. Premises:<br>1️⃣ All cats are animals.<br>2️⃣ No animal is a plant.<br>3️⃣ All plants are living things.<br><br>Which of the following conclusions definitely follows?<br>(A) No cat is a plant.<br>(B) Some cats are living things.<br>(C) All animals are plants.<br>(D) Some living things are animals.
medium
A. No cat is a plant.
B. Some cats are living things.
C. All animals are plants.
D. Some living things are animals.

Solution

  1. Step 1: Link First Two Premises

    All cats ⊂ animals, no animal ⊂ plant ⇒ No cat ⊂ plant.

  2. Step 2: Validate

    This follows directly and necessarily.

  3. Final Answer:

    No cat is a plant → Option A

  4. Quick Check:

    Negative premise extends across category ✅
Hint: ‘No A is B’ transfers exclusion to all subgroups.
Common Mistakes: Forgetting negative statements propagate exclusion.
5. Premises:<br>1️⃣ All engineers are logical.<br>2️⃣ Some logical people are mathematicians.<br>3️⃣ No mathematician is lazy.<br><br>Which of the following conclusions definitely follows?<br>(A) Some engineers are mathematicians.<br>(B) Some logical people are not lazy.<br>(C) All engineers are lazy.<br>(D) No logical person is an engineer.
medium
A. Some engineers are mathematicians.
B. Some logical people are not lazy.
C. All engineers are lazy.
D. No logical person is an engineer.

Solution

  1. Step 1: Combine Premises

    Some logical ⊂ mathematicians; no mathematician ⊂ lazy ⇒ Some logical ⊂ not lazy.

  2. Step 2: Evaluate Options

    ‘Some logical people are not lazy’ directly follows ⇒ ✅ True.

  3. Final Answer:

    Some logical people are not lazy → Option B

  4. Quick Check:

    Logical → Mathematician → Not Lazy ✅
Hint: When a negative appears at the end, carry exclusion backward.
Common Mistakes: Linking engineers with mathematicians unnecessarily.

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