Theorems in propositional logic | Rules of inference

Material implication (rule of inference)

In propositional logic, material implication is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that P implies Q is logically equivalent to not- or and that either form can replace the other in logical proofs. In other words, if is true, then must also be true, while if is not true, then cannot be true either; additionally, when is not true, may be either true or false. Where "" is a metalogical symbol representing "can be replaced in a proof with," and P and Q are any given logical statements. To illustrate this, consider the following statements: * : Sam ate an orange for lunch * : Sam ate a fruit for lunch Then, to say, "Sam ate an orange for lunch" implies "Sam ate a fruit for lunch". Logically, if Sam did not eat a fruit for lunch, then Sam also cannot have eaten an orange for lunch (by contraposition). However, merely saying that Sam did not eat an orange for lunch provides no information on whether or not Sam ate a fruit (of any kind) for lunch. (Wikipedia).

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Thanks so much for watching! Please comment below on what topics you'd like to see covered next!

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Related pages

Propositional calculus | Contraposition | Formal proof | Symbol (formal) | Metalogic | Rule of replacement | Negation | Logical equivalence | Law of excluded middle | Truth table | Validity (logic) | Logical disjunction | Material conditional | Antecedent (logic)