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Deduction by Contradiction

Introduction

Deduction by Contradiction-ல், ஒரு conclusion உண்மையா என்பதை சோதிக்க, அதன் எதிர்மறை (opposite) யை assume செய்து, அந்த assumption ஒரு contradiction-க்கு வழிவகுக்கிறதா என்பதை பார்க்கிறோம். direct deduction எளிதாக இல்லாத போது, conclusions-ஐ மறைமுகமாக (indirectly) உறுதிப்படுத்த இந்த reasoning technique உதவுகிறது.

இது analytical reasoning-ல் பயன்படும் முக்கியமான logical pattern; எதிர்மறை assumption தோல்வியடைகிறதே என்றால், conclusion கட்டாயமாக true ஆகவேண்டும் என்பதைக் நிரூபிக்கிறது.

Pattern: Deduction by Contradiction

Pattern

முக்கிய கருத்து: Conclusion-ன் opposite-ஐ assume செய்யவும். அது contradiction-க்கு வழிவகுத்தால், conclusion true ஆகவேண்டும்.

இந்த indirect reasoning pattern, logical negation மூலம் தவறான assumptions-ஐ நீக்குகிறது.

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.

Conclusion-க்கு சரியான logical evaluation எது?
(A) Conclusion invalid.
(B) Conclusion directly follows.
(C) Conclusion follows by contradiction.
(D) Conclusion cannot be determined.

Solution

  1. Step 1: Symbolically எழுதவும்

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

  2. Step 2: Opposite-ஐ assume செய்யவும்

    Conclusion false என்று assume செய்கிறோம் - அதாவது மழை பெய்தது (R true).

  3. Step 3: Rule-ஐ apply செய்யவும்

    If R → W மற்றும் R true ⇒ W true ஆகவேண்டும். ஆனால் given-ல் ¬W. இது contradiction.

  4. Step 4: Deduce

    ஆகையால் “மழை பெய்தது” என்ற assumption செல்லாது. எனவே “மழை பெய்யவில்லை” என்பது true.

  5. Final Answer:

    Conclusion contradiction மூலம் follow ஆகிறது → Option C

  6. Quick Check:

    Opposite-ஐ assume செய்தால் logical conflict வருகிறது ⇒ original conclusion valid ✅

Quick Variations

1️⃣ Direct deduction தெளிவாக இல்லாத போது பயன்படுத்தப்படும்.

2️⃣ “If A → B மற்றும் ¬B ⇒ ¬A” போன்ற conditional negations உள்ள கேள்விகளில் பொதுவாக வரும்.

3️⃣ Assumptions-ன் impossibility அல்லது falsity-ஐ நிரூபிக்கவும் பயன்படுத்தப்படுகிறது.

Trick to Always Use

  • Step 1: Conclusion-ன் negation-ஐ assume செய்யவும்.
  • Step 2: Given premises-உடன் consistency உள்ளதா என சரிபார்க்கவும்.
  • Step 3: Contradiction வந்தால், negated assumption false - அதனால் conclusion true.

Summary

Summary

  • முதலில் conclusion-ன் opposite-ஐ assume செய்யவும்.
  • Given premises-ஐ logically apply செய்து contradiction வருகிறதா பார்க்கவும்.
  • Contradiction வந்தால், assumption false; conclusion true.
  • இது logical deduction problems-ல் பயன்படுத்தப்படும் indirect proof method.

நினைவில் வைக்க வேண்டிய example:
If R → W மற்றும் ¬W, then ¬R (R-ஐ assume செய்தால் contradiction வரும்).

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.

Mock Test

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