Material implication can also be characterized inferentially by modus ponens, modus tollens, conditional proof, and classical reductio ad absurdum.
It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many programming languages.
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
In Arithmetices Principia: Nova Methodo Exposita (1889), Peano expressed the proposition "If
[1] Russell followed Peano in his Principia Mathematica (1910–1913), in which he expressed the proposition "If A, then B" as
[6] From a classical semantic perspective, material implication is the binary truth functional operator which returns "true" unless its first argument is true and its second argument is false.
: The logical cases where the antecedent A is false and A → B is true, are called "vacuous truths".
Examples are ... Material implication can also be characterized deductively in terms of the following rules of inference.
[citation needed] Unlike the semantic definition, this approach to logical connectives permits the examination of structurally identical propositional forms in various logical systems, where somewhat different properties may be demonstrated.
For example, in intuitionistic logic, which rejects proofs by contraposition as valid rules of inference,
is not a propositional theorem, but the material conditional is used to define negation.
[clarification needed] When disjunction, conjunction and negation are classical, material implication validates the following equivalences: Similarly, on classical interpretations of the other connectives, material implication validates the following entailments: Tautologies involving material implication include: Material implication does not closely match the usage of conditional sentences in natural language.
For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd, then 3 is prime" is typically judged false.
Similarly, any material conditional with a true consequent is itself true, but speakers typically reject sentences such as "If I have a penny in my pocket, then Paris is in France".
These classic problems have been called the paradoxes of material implication.
[7] In addition to the paradoxes, a variety of other arguments have been given against a material implication analysis.
For instance, counterfactual conditionals would all be vacuously true on such an account, when in fact some are false.
[8] In the mid-20th century, a number of researchers including H. P. Grice and Frank Jackson proposed that pragmatic principles could explain the discrepancies between natural language conditionals and the material conditional.
On their accounts, conditionals denote material implication but end up conveying additional information when they interact with conversational norms such as Grice's maxims.
[7][9] Recent work in formal semantics and philosophy of language has generally eschewed material implication as an analysis for natural-language conditionals.
[7] Thus semantic analyses of conditionals typically propose alternative interpretations built on foundations such as modal logic, relevance logic, probability theory, and causal models.
[9][7][10] Similar discrepancies have been observed by psychologists studying conditional reasoning, for instance, by the notorious Wason selection task study, where less than 10% of participants reasoned according to the material conditional.
Some researchers have interpreted this result as a failure of the participants to conform to normative laws of reasoning, while others interpret the participants as reasoning normatively according to nonclassical laws.