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Chapter: Logical Arguments Used in Mathematical Proofs

In the following, we will introduce some most important types of valid logical arguments. While introducing them, we will also mathematically prove that they are indeed valid. You will learn the techniques of the following types of valid logical arguments:

• deductive reasoning,
• modus ponens,
• modus tollens,
• hypothetical syllogism,
• disjunctive syllogism,
• the proving principle by contradiction,
• the proving principle by contraposition,
• the proving principle by induction. Feel free to use these techniques, not only when writing mathematical proofs, but also in daily life, e.g. when trying to convince your audience or discussion partners.

Table of Contents

Explanations: 1

1. Lemma: A Criterion for Valid Logical Arguments
2. Lemma: Modus Ponens
3. Lemma: Modus Tollens
4. Lemma: Hypothetical Syllogism
5. Lemma: Disjunctive Syllogism
6. Lemma: The Proving Principle by Contradiction
7. Lemma: The Proving Principle By Contraposition, Contrapositive
8. Lemma: The Proving Principle by Complete Induction
9. Lemma: The Proving Principle by Transfinite Induction

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References

Bibliography

1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016