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The paradoxes of material implication are a group of formulae that are truths of classical logic but are intuitively problematic. The root of the paradoxes lies in a mismatch between the interpretation of the validity of logical implication in natural language, and its formal interpretation in classical logic, dating back to George Boole's algebraic logic. In classical logic, implication describes conditional if-then statements using a truth-functional interpretation, i.e. "p implies q" is defined to be "it is not the case that p is true and q false". Also, "p implies q" is equivalent to "p is false or q is true". For example, "if it is raining, then I will bring an umbrella", is equivalent to "it is not raining, or I will bring an umbrella, or both". This truth-functional interpretation of implication is called material implication or material conditional. The paradoxes are logical statements which are true but whose truth is intuitively surprising to people who are not familiar with them. If the terms 'p', 'q' and 'r' stand for arbitrary propositions then the main paradoxes are given formally as follows: # , p and its negation imply q. This is the ''paradox of entailment''. # , if p is true then it is implied by every q. # , if p is false then it implies every q. This is referred to as 'explosion'. In these cases, the statement is said to be vacuously true. # , either q or its negation is true, so their disjunction is implied by every p. # , if p, q and r are three arbitrary propositions, then either p implies q or q implies r. This is because if q is true then p implies it, and if it is false then q implies any other statement. Since r can be p, it follows that given two arbitrary propositions, one must imply the other, even if they are mutually contradictory. For instance, "Nadia is in Barcelona implies Nadia is in Madrid, or Nadia is in Madrid implies Nadia is in Barcelona." This truism sounds like nonsense in ordinary discourse. # , if p does not imply q then p is true and q is false. NB if p were false then it would imply q, so p is true. If q were also true then p would imply q, hence q is false. This paradox is particularly surprising because it tells us that if one proposition does not imply another then the first is true and the second false. The paradoxes of material implication arise because of the truth-functional definition of material implication, which is said to be true merely because the antecedent is false or the consequent is true. By this criterion, "If the moon is made of green cheese, then the world is coming to an end," is true merely because the moon isn't made of green cheese. By extension, any contradiction implies anything whatsoever, since a contradiction is never true. (All paraconsistent logics must, by definition, reject (1) as false.) Also, any tautology is implied by anything whatsoever, since a tautology is always true. To sum up, although it is deceptively similar to what we mean by "logically follows" in ordinary usage, material implication does not capture the meaning of "if... then". ==Paradox of entailment== 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 :''Water exists.'' 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, as it suggests that the above is a valid argument. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「paradoxes of material implication」の詳細全文を読む スポンサード リンク
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