Introduction To Logic - Week1 - Part 01

 

What is Logic?

         Logic is the systematic study of principles of valid reasoning and inference. It provides rules and techniques to distinguish correct reasoning from incorrect reasoning. In simple terms, logic helps us determine whether an argument or statement is true or false based on given premises.

In Conclusion, Logic is a fundamental discipline that helps us reason correctly and draw valid conclusions based on given information. It is widely used in mathematics, philosophy, computer science, and artificial intelligence to ensure precise and structured thinking.

Practice Quiz: Need for Formal Logics in Computer Science

Q1. Observe the statement: “Mathematical logic is about inferring ____(X)____ from the given ____(Y)____.” Fill in the blanks (X) and (Y) by selecting the most appropriate option.

Ans. X = Conclusions; Y = Premises

In Depth Soln.

Breaking Down the Concept

Mathematical logic is a branch of mathematics that deals with reasoning, proof, and the principles of valid inference. The key idea is that we derive conclusions (results) based on premises (assumptions or given information).

• Premises (Y) are statements or facts that are assumed to be true.
• Conclusions (X) are logical deductions that follow from the premises.

For example, consider a simple logical argument:

1. Premise: All humans are mortal.
2. Premise: Socrates is a human.
3. Conclusion: Therefore, Socrates is mortal.

This follows the logical reasoning process where conclusions are inferred from premises.

X = Conclusions; Y = Premises ✅ (Correct)

• This correctly matches the idea of mathematical logic. We derive conclusions based on given premises.

Conclusion

Mathematical logic is the scientific study of how conclusions can be logically derived from given premises.

Q2. Which of the following problems has had a long history of false proofs and was finally proved using a theorem which is based on mathematical logic?

(a) Satisfiability of propositional formulae

(b) Four-coloring

(c) Matrix multiplication

(d) Prime factorization

Ans. (b)  Four-coloring

In Depth Soln.

Breaking Down the Concept

The Four-Color Theorem states that:
"Any map can be colored using at most four colors in such a way that no two adjacent regions share the same color."

Source: Wikipedia

This problem has a long history of incorrect proofs and was one of the first major theorems to be proved using computers.

• False Proofs: Many mathematicians attempted to prove this theorem for over a century, but most early proofs were incorrect.

• Final Proof (1976): Mathematicians Kenneth Appel and Wolfgang Haken used mathematical logic and computer-assisted proof to solve the problem. Their proof checked 1,936 cases using a computer, making it one of the first major theorems to rely on computation.

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Evaluating the Answer Choices

1. Matrix Multiplication ❌ (Incorrect)

   • Matrix multiplication is a well-established mathematical operation. It does not have a history of false proofs and was not proved using logic-based theorems.

2. Four-Coloring ✅ (Correct)

   • The Four-Color Theorem has a long history of incorrect proofs and was finally proven using mathematical logic and computational methods.

3. Prime Factorization ❌ (Incorrect)

   • Prime factorization is a fundamental concept in number theory and is not known for false proofs. While it is important in cryptography, it does not fit the context of this question.

4. Satisfiability of Propositional Formulae ❌ (Incorrect)

   • This relates to Boolean logic and computational complexity (e.g., SAT problems), but it does not have a long history of false proofs like the Four-Color Theorem.

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Conclusion

The correct answer is:
✔ Four-Coloring

The Four-Color Theorem is historically significant because of its many incorrect proofs and its final proof using mathematical logic and computers in 1976.

Question 2Which of the following problems has had a long history of false proofs and was finally proved using a theorem which is based on mathematical logic?