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Deduction with Multiple Premises

Introduction

In logical reasoning, many problems require combining two or more premises to reach a valid conclusion. These questions test your ability to link statements step by step - forming a logical chain across multiple premises.

This pattern is important because it develops transitive reasoning and helps in solving complex syllogisms and inference-based sets.

Pattern: Deduction with Multiple Premises

Pattern

The key idea is to combine given premises (facts) logically - where one statement leads to another - and derive a new conclusion that follows necessarily.

For example, if A → B and B → C, then we can validly conclude A → C. This is called Transitive Deduction.

Step-by-Step Example

Question

Premises:
1️⃣ All fruits are food.
2️⃣ All apples are fruits.
3️⃣ All food items are edible.

Which conclusion definitely follows?
(A) All fruits are apples.
(B) All apples are edible.
(C) Some apples are not edible.
(D) All edible items are apples.

Solution

  1. Step 1: Link the first two premises

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

  2. Step 2: Add the third premise

    All food ⊂ edible ⇒ All apples ⊂ edible.

  3. Step 3: Verify other options

    ‘All fruits are apples’ ❌ (reverse), ‘Some apples not edible’ ❌ (contradicts given), ‘All edible are apples’ ❌ (too broad).

  4. Final Answer:

    All apples are edible → Option B

  5. Quick Check:

    Chain - Apples → Fruits → Food → Edible ✅

Quick Variations

1. Transitive deduction with 3 or more premises.

2. Mixed premises containing negatives (e.g., “No A is B, All B are C”).

3. Indirect or reversed logical chains.

4. Conditional deductions involving “if-then” forms across multiple links.

Trick to Always Use

  • Step 1: Arrange premises in order (A → B → C).
  • Step 2: Link common terms to form a logical chain.
  • Step 3: Test for direction - only valid in forward logical flow.
  • Step 4: Reject reverse or unrelated statements immediately.

Summary

Summary

  • Multiple premises can be combined only through shared terms.
  • Transitive logic works when all links connect in the same direction.
  • Negatives break the direct chain - handle with care.
  • Always read from the first subject to the final object of relation.

Example to remember:
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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