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In logic, the material conditional (sometimes called the implies operator) is a binary truth function ⊃ from truth-values to truth-values. In symbols, a (material) conditional is written as one of the following: The material conditional is false when X is true and Y is false - otherwise, it is true. (Here, X and Y are formulas of the language of a logic that denote truth-values and the symbol '⊃' is used as a name for itself.) We call the first argument, i.e. X, the antecedent of the conditional, and Y the consequent. The material conditional is also commonly referred to as material implication and it is said that, in (1), X (materially) implies Y. A close approximation to the material conditional is the English construction 'if...then...', where the ellipses are to be filled with English sentences.
Formal properties The material conditional is not to be confused with the entailment relation ⊨ (which is used here as a name for itself). But there is a close relationship between the two in most logics, including classical logic which we only consider here. For example, the following principles hold: These principles do not hold in all logics, however. Philosophical problems with material conditional The use of the operator is stipulated by logicians, and, as a result, can yield some unexpected truths in natural language. For example, any material conditional statement with a false antecedent is true. So the statement "2 is odd implies 2 is even" is true. Similarly, any material conditional with a true consequent is true. So the statement, "If pigs fly, then Paris is in France" is true. These problems are known as the paradoxes of material implication, though they are not really paradoxes in the strict sense. These unexpected truths arise because speakers of English (and other natural languages) are tempted to equivocate between the material conditional and the indicative conditional, or other conditional statements, like the counterfactual conditional and the material biconditional. It is not surprising that a rigorously defined truth-functional operator does not correspond exactly to all notions of implication or otherwise expressed by 'if...then...' sentences in English (or their equivalents in other natural languages). For an overview of some the various analyses, formal and informal, of conditionals, see the "References" section below. Logical operators Related topics | ||||||||
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