Paradoxes of material implication

The paradoxes of material implication are a group of true formulae involving material conditionals whose translations into natural language are intuitively false when the conditional is translated as "if ... then ...". A material conditional formula is true unless is true and is false. If natural language conditionals were understood in the same way, that would mean that the sentence "If the Nazis had won World War Two, everybody would be happy" is vacuously true. Given that such problematic consequences follow from a seemingly correct assumption about logic, they are called paradoxes. They demonstrate a mismatch between classical logic and robust intuitions about meaning and reasoning. As the best known of the paradoxes, and most formally simple, the paradox of entailment makes the best introduction. In natural language, an instance of the paradox of entailment arises: It is raining And It is not raining Therefore George Washington is made of rakes. This arises from the principle of explosion, a law of classical logic stating that inconsistent premises always make an argument valid; that is, inconsistent premises imply any conclusion at all. This seems paradoxical because although the above is a logically valid argument, it is not sound (not all of its premises are true). Validity is defined in classical logic as follows: An argument (consisting of premises and a conclusion) is valid if and only if there is no possible situation in which all the premises are true and the conclusion is false. For example a valid argument might run: If it is raining, water exists (1st premise) It is raining (2nd premise) Water exists (Conclusion) In this example there is no possible situation in which the premises are true while the conclusion is false. Since there is no counterexample, the argument is valid. But one could construct an argument in which the premises are inconsistent. This would satisfy the test for a valid argument since there would be no possible situation in which all the premises are true and therefore no possible situation in which all the premises are true and the conclusion is false.