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Concept
Material conditional
Summary
The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol is interpreted as material implication, a formula is true unless is true and is false. Material implication can also be characterized inferentially by modus ponens, modus tollens, conditional proof, and classical reductio ad absurdum.
Material implication is used in all the basic systems of classical logic as well as some nonclassical logics. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many programming languages. However, many logics replace material implication with other operators such as the strict conditional and the variably strict conditional. Due to the paradoxes of material implication and related problems, material implication is not generally considered a viable analysis of conditional sentences in natural language.
In logic and related fields, the material conditional is customarily notated with an infix operator . The material conditional is also notated using the infixes and . In the prefixed Polish notation, conditionals are notated as . In a conditional formula , the subformula is referred to as the antecedent and is termed the consequent of the conditional. Conditional statements may be nested such that the antecedent or the consequent may themselves be conditional statements, as in the formula .
In Arithmetices Principia: Nova Methodo Exposita (1889), Peano expressed the proposition “If then ” as Ɔ with the symbol Ɔ, which is the opposite of C. He also expressed the proposition as Ɔ . Hilbert expressed the proposition “If A then B” as in 1918. Russell followed Peano in his Principia Mathematica (1910–1913), in which he expressed the proposition “If A then B” as . Following Russell, Gentzen expressed the proposition “If A then B” as . Heyting expressed the proposition “If A then B” as at first but later came to express it as with a right-pointing arrow. Bourbaki expressed the proposition “If A then B” as in 1954.
Official source
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