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L ogic A term used by Russell and Whitehead , representing the truth-function of two propositions P and Q in the form of the statement “If P then Q.” The relation is symbolized by a horseshoe ⊃, written as “P → Q,” or alternatively by an arrow →, written as “P → Q.” It is true that P materially implies Q in each of the following three cases: (1) both P and Q are true; (2) P is false, and Q is true; (3) both P and Q are false. It is false only if P is true and Q is false. This is called material implication because what is expressed by the sign ‘⊃’ is different from our ordinary notion of implication . A statement such as “If Rome is in Italy, then London is beautiful” is true by material implication, but does not seem to be an implication at all in the ordinary sense, for there is no relation between antecedent and consequent. It is due to this difference that material implication leads to many paradoxes . Some philosophers claim therefore that we should say that it is a material conditional relation instead of a relation of material implication. “The relation in virtue of which it is possible for us validly to infer is what I call material implication … The relation holds, in fact, when it does hold, without any reference to the truth or false-hood of the proposition involved.” Russell, The Principles of Mathematic ... log in or subscribe to read full text