Logical Arguments & Validity

Math for CS · Axiom Academy

Modus ponens, modus tollens, syllogisms, and formal proofs Validity is about structure , not about whether the premises are actually true in real life. A valid argument with false premises can have a false conclusion — but a valid argument with true premises must have a true conclusion. "If it rains, the ground is wet. It is raining. Therefore the ground is wet." This is the single most common inference rule and the basis of if-then execution in programming. "If it rains, the ground is wet. The ground is NOT wet. Therefore it is NOT raining." Watch out for these common fallacies (invalid argument forms): A formal proof chains inference rules together. Each line is justified by a rule applied to previous lines. Modus ponens ( p q , p , therefore q ) and modus tollens ( p q , q , therefore p ) are the two fundamental inference rules. Affirming the consequent and denying the antecedent are the two classic fallacies. Formal proofs chain valid rules step by step — each line cites its justification. Resolution is the rule that powers automated theorem provers and SAT solvers.

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