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• Paradoxes_of_material_implication abstract "The paradoxes of material implication are a group of formulas which are truths of classical logic, but which are intuitively problematic. One of these paradoxes is the paradox of entailment.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'. , 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".".
• Paradoxes_of_material_implication wikiPageID "12857474".
• Paradoxes_of_material_implication wikiPageRevisionID "583814051".
• Paradoxes_of_material_implication hasPhotoCollection Paradoxes_of_material_implication.
• Paradoxes_of_material_implication id "p/s090470".
• Paradoxes_of_material_implication title "Strict implication calculus".
• Paradoxes_of_material_implication subject Category:Logical_consequence.
• Paradoxes_of_material_implication subject Category:Paradoxes.
• Paradoxes_of_material_implication type Abstraction100002137.
• Paradoxes_of_material_implication type Communication100033020.
• Paradoxes_of_material_implication type Contradiction107206887.
• Paradoxes_of_material_implication type Falsehood106756407.
• Paradoxes_of_material_implication type Message106598915.
• Paradoxes_of_material_implication type Paradox106724559.
• Paradoxes_of_material_implication type Paradoxes.
• Paradoxes_of_material_implication type Statement106722453.
• Paradoxes_of_material_implication comment "The paradoxes of material implication are a group of formulas which are truths of classical logic, but which are intuitively problematic. One of these paradoxes is the paradox of entailment.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.".
• Paradoxes_of_material_implication label "Paradojas de la implicación material".
• Paradoxes_of_material_implication label "Paradoxes of material implication".
• Paradoxes_of_material_implication label "Paradoxien der materialen Implikation".
• Paradoxes_of_material_implication label "Paradoxos da implicação material".
• Paradoxes_of_material_implication label "Парадокс импликации".
• Paradoxes_of_material_implication sameAs Paradoxien_der_materialen_Implikation.
• Paradoxes_of_material_implication sameAs Paradojas_de_la_implicación_material.
• Paradoxes_of_material_implication sameAs Paradoxos_da_implicação_material.
• Paradoxes_of_material_implication sameAs m.02x7_47.
• Paradoxes_of_material_implication sameAs Q1848046.
• Paradoxes_of_material_implication sameAs Q1848046.
• Paradoxes_of_material_implication sameAs Paradoxes_of_material_implication.
• Paradoxes_of_material_implication wasDerivedFrom Paradoxes_of_material_implication?oldid=583814051.
• Paradoxes_of_material_implication isPrimaryTopicOf Paradoxes_of_material_implication.