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Deduction by Contradiction

Introduction

In Deduction by Contradiction, we test the truth of a conclusion by assuming the opposite of it and proving that this assumption leads to a contradiction. This reasoning technique helps confirm conclusions indirectly when direct deduction isn’t simple.

It’s an essential logical pattern used in analytical reasoning to prove necessity - if the opposite assumption fails, the conclusion must hold true.

Pattern: Deduction by Contradiction

Pattern

The key idea is: Assume the opposite of the conclusion. If that leads to a contradiction, the conclusion must be true.

This indirect reasoning pattern eliminates false assumptions through logical negation.

Step-by-Step Example

Question

Statements:
1️⃣ If it rains, the ground becomes wet.
2️⃣ The ground is not wet.
Conclusion: It did not rain.

Choose the correct logical evaluation of the conclusion:
(A) The conclusion is invalid.
(B) The conclusion follows directly.
(C) The conclusion follows by contradiction.
(D) The conclusion cannot be determined.

Solution

  1. Step 1: Express symbolically

    Let R = “It rains” and W = “Ground is wet”. Given: If R → W, and ¬W (not wet).

  2. Step 2: Assume opposite

    Assume the conclusion is false - suppose it did rain (R true).

  3. Step 3: Apply rule

    If R → W, and R is true ⇒ W must be true. But given ¬W. This is a contradiction.

  4. Step 4: Deduce

    Therefore, the assumption “It rained” cannot hold. Hence, “It did not rain” must be true.

  5. Final Answer:

    The conclusion follows by contradiction → Option C

  6. Quick Check:

    Assuming the opposite creates logical conflict ⇒ original conclusion is valid ✅

Quick Variations

1️⃣ Used when direct deduction isn’t clear.

2️⃣ Common in problems involving conditional negations like “If A → B and ¬B ⇒ ¬A”.

3️⃣ Also used to prove impossibility or falsity of assumptions.

Trick to Always Use

  • Step 1: Assume the negation of the conclusion.
  • Step 2: Check consistency with given premises.
  • Step 3: If contradiction occurs, the negated assumption is false - so the conclusion is true.

Summary

Summary

  • Assume the opposite of the conclusion first.
  • Apply given premises logically to check if contradiction arises.
  • If contradiction appears, the assumption is false and conclusion must be true.
  • This is an indirect proof method used in logical deduction problems.

Example to remember:
If R → W and ¬W, then ¬R (since assuming R causes contradiction).

Practice

(1/5)
1. Statements:<br>1️⃣ If the shop is open, customers can buy goods.<br>2️⃣ Customers could not buy goods today.<br><br>Conclusions:<br>I. The shop was not open today.<br>II. Customers did not have money.<br><br>Which conclusion definitely follows?
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: Identify condition

    If Shop Open → Customers Can Buy.

  2. Step 2: Given

    Customers could not buy goods ⇒ effect didn’t happen.

  3. Step 3: Apply contradiction

    Assume shop was open ⇒ they should buy. Contradiction arises. Hence, shop was not open.

  4. Final Answer:

    Only Conclusion I follows → Option A

  5. Quick Check:

    Negating cause matches given effect ✅
Hint: When effect fails, check if cause must be false using contradiction.
Common Mistakes: Assuming external reasons (like lack of money) without data.
2. Statements:<br>1️⃣ If it rains, the picnic will be cancelled.<br>2️⃣ The picnic was not cancelled.<br><br>Conclusions:<br>I. It did not rain.<br>II. The picnic continued as planned.<br><br>Which conclusion definitely follows?
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: Condition

    If Rain → Picnic Cancelled.

  2. Step 2: Given

    Picnic not cancelled ⇒ ¬Cancelled.

  3. Step 3: Contradiction check

    If it had rained, picnic must cancel → contradiction. So, it did not rain, and picnic continued.

  4. Final Answer:

    Both I and II follow → Option C

  5. Quick Check:

    No rain ⇒ no cancellation ✅
Hint: When effect didn’t occur, assume cause’s negation logically holds.
Common Mistakes: Overlooking the positive inference ('picnic continued').
3. Statements:<br>1️⃣ If a student studies well, they pass the exam.<br>2️⃣ Ravi failed the exam.<br><br>Conclusions:<br>I. Ravi did not study well.<br>II. Ravi studied well but was unlucky.<br><br>Which conclusion definitely follows?
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: Condition

    If Study Well → Pass Exam.

  2. Step 2: Given

    Ravi failed ⇒ ¬Pass.

  3. Step 3: Contradiction test

    Assume Ravi studied well ⇒ he must pass. Contradiction. So, he did not study well.

  4. Final Answer:

    Only Conclusion I follows → Option A

  5. Quick Check:

    Negating effect proves negation of cause ✅
Hint: Failure of effect disproves sufficiency of cause.
Common Mistakes: Adding emotional factors (luck, difficulty) not in logic.
4. Statements:<br>1️⃣ If the alarm rings, everyone will wake up.<br>2️⃣ Everyone was sleeping.<br><br>Conclusions:<br>I. The alarm did not ring.<br>II. Everyone ignored the alarm.<br><br>Which conclusion definitely follows?
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: Logic

    If Alarm Rings → All Wake.

  2. Step 2: Given

    All sleeping ⇒ ¬Wake.

  3. Step 3: Contradiction check

    If alarm had rung ⇒ all must wake (contradiction). So, alarm didn’t ring.

  4. Final Answer:

    Only Conclusion I follows → Option A

  5. Quick Check:

    Negative effect negates cause ✅
Hint: Apply contrapositive reasoning: If no effect, no cause.
Common Mistakes: Assuming external reasons like ignoring alarm.
5. Statements:<br>1️⃣ If the lights are on, electricity is available.<br>2️⃣ Electricity is not available.<br><br>Conclusions:<br>I. The lights are off.<br>II. The lights are on but not working.<br><br>Which conclusion definitely follows?
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: Condition

    If Lights On → Electricity Available.

  2. Step 2: Given

    No electricity ⇒ ¬Available.

  3. Step 3: Contradiction test

    If lights were on ⇒ electricity must exist (contradiction). Hence, lights are off.

  4. Final Answer:

    Only Conclusion I follows → Option A

  5. Quick Check:

    Contrapositive valid: ¬Electricity ⇒ ¬Lights ✅
Hint: Always match given negation to cause through contrapositive logic.
Common Mistakes: Inventing false scenarios like malfunctioning devices.

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