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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.

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Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related. They may be viewed as a family of substructural or modal logics. It is generally, but not universally, called relevant logic by British and, especially, Australian logicians, and relevance logic by American logicians. Relevance logic aims to capture aspects of implication that are ignored by the "material implication" operator in classical truth-functional logic, namely the notion of relevance between antecedent and conditional of a true implication.

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Counterfactual conditionals (also subjunctive or X-marked) are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactuals are contrasted with indicatives, which are generally restricted to discussing open possibilities. Counterfactuals are characterized grammatically by their use of fake tense morphology, which some languages use in combination with other kinds of morphology including aspect and mood.

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In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. It is sometimes said that a statement is vacuously true because it does not really say anything. For example, the statement "all cell phones in the room are turned off" will be true when no cell phones are in the room.

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