Direct Proofs

Mathematical Logic · Axiom Academy

Learn the fundamental technique of direct proof: starting from premises and systematically deriving conclusions using valid inference rules. Think of a direct proof as building a bridge from what you know to what you want to prove. Each plank of the bridge must be solidly supported by the previous ones or by the ground (the premises). 2. Structure of a Direct Proof Premises: State what is given or assumed Inference Steps: Apply logical rules to derive new statements Conclusion: Reach the desired result Each inference step must cite a valid rule. Common rules include Modus Ponens, Modus Tollens, Hypothetical Syllogism, Conjunction, Simplification, and Addition. 3. Example: Basic Direct Proof We'll build the proof step by step, showing how each new statement follows from previous ones using valid inference rules. 4. Proving Implications: The Direct Method Assume the antecedent p is true Use logical reasoning to derive the consequent q This is the most common form of mathematical proof. When we want to prove "if p then q", we start by assuming p and show that q must follow. This mirrors the definition of implication: if p is true, then q must also be true. 5. Conditional Proof (→-Introduction) Conditional proof is a powerful technique. We temporarily assume the antecedent, derive the consequent within that assumption, then "discharge" the assumption to conclude the full implication. This is also called →-introduction or the deduction theorem. 6. Example: Using Conditional Proof

This is the written version of the interactive lesson above. See the full Mathematical Logic course.