Modus Ponens and Modus Tollens

Mathematical Logic · Axiom Academy

LESSON Modus Ponens and Modus Tollens Master the fundamental rules of logical inference that form the foundation of mathematical proof Mathematical Logic • Unit 3 - Proof Theory 1. Modus Ponens: The Rule of Affirmation Modus Ponens (Latin for "mode of affirmation") is a fundamental inference rule. If we know that "p implies q" is true, and we also know that "p" is true, then we can conclude that "q" must be true. 2. Why Modus Ponens is Valid: Truth Table Proof Let's prove that Modus Ponens is a valid inference rule using a truth table. We need to show that whenever both premises are true, the conclusion must also be true. Let's see how Modus Ponens works in both formal logic and natural language examples. Rule: If it is raining, then the ground is wet. Conclusion: The ground is wet. Identify the conditional statement (p → q) Verify that the antecedent (p) is true Conclude that the consequent (q) is true 4. Modus Tollens: The Rule of Denial Modus Tollens (Latin for "mode of denial") is another fundamental inference rule. If we know that "p implies q" is true, and we know that "q" is false, then we can conclude that "p" must be false. 5. Why Modus Tollens is Valid: Truth Table Proof Let's prove that Modus Tollens is valid by showing that whenever both premises are true, the conclusion must be true. Let's explore how Modus Tollens works in practice with concrete examples. Conclusion: It is NOT raining. Verify that the consequent (q) is false

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