Material conditional

The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol $$\rightarrow$$ is interpreted as material implication, a formula $$ P \rightarrow Q$$ is true unless $$P$$ is true and $$Q$$ 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.

Notation
In logic and related fields, the material conditional is customarily notated with an infix operator $$\to$$. The material conditional is also notated using the infixes $$\supset$$ and $$\Rightarrow$$. In the prefixed Polish notation, conditionals are notated as $$Cpq$$. In a conditional formula $$p\to q$$, the subformula $$p$$ is referred to as the antecedent and $$q$$ 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 $$(p\to q)\to(r\to s)$$.

History
In Arithmetices Principia: Nova Methodo Exposita (1889), Peano expressed the proposition "If $$A$$, then $$B$$" as $$A$$ Ɔ $$B$$ with the symbol Ɔ, which is the opposite of C. He also expressed the proposition $$A\supset B$$ as $$A$$ Ɔ $$B$$. Hilbert expressed the proposition "If A, then B" as $$A\to B$$ in 1918. Russell followed Peano in his Principia Mathematica (1910–1913), in which he expressed the proposition "If A, then B" as $$A\supset B$$. Following Russell, Gentzen expressed the proposition "If A, then B" as $$A\supset B$$. Heyting expressed the proposition "If A, then B" as $$A\supset B$$ at first but later came to express it as $$A\to B$$ with a right-pointing arrow. Bourbaki expressed the proposition "If A, then B" as $$A\Rightarrow B$$ in 1954.

Semantics
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. This semantics can be shown graphically in a truth table such as the one below. One can also consider the equivalence $$A \to B \equiv \neg (A \land \neg B) \equiv \neg A \lor B$$.

Truth table
The truth table of $$A \rightarrow B$$:

The logical cases where the antecedent $A$ is false and $A → B$ is true, are called "vacuous truths". Examples are ...
 * ... with $B$ false: "If Marie Curie is a sister of Galileo Galilei, then Galileo Galilei is a brother of Marie Curie",
 * ... with $B$ true: "If Marie Curie is a sister of Galileo Galilei, then Marie Curie has a sibling.".

Deductive definition
Material implication can also be characterized deductively in terms of the following rules of inference.


 * Modus ponens
 * Conditional proof
 * Classical contraposition
 * Classical reductio ad absurdum

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, $$(A \to B) \Rightarrow \neg A \lor B $$ is not a propositional theorem, but the material conditional is used to define negation.

Formal properties
When disjunction, conjunction and negation are classical, material implication validates the following equivalences:
 * Contraposition: $$P \to Q \equiv \neg Q \to \neg P$$
 * Import-export: $$P \to (Q \to R) \equiv (P \land Q) \to R$$
 * Negated conditionals: $$\neg(P \to Q) \equiv P \land \neg Q$$
 * Or-and-if: $$P \to Q \equiv \neg P \lor Q$$
 * Commutativity of antecedents: $$\big(P \to (Q \to R)\big) \equiv \big(Q \to (P \to R)\big)$$
 * Left distributivity: $$\big(R \to (P \to Q)\big) \equiv \big((R \to P) \to (R \to Q)\big)$$

Similarly, on classical interpretations of the other connectives, material implication validates the following entailments:
 * Antecedent strengthening: $$P \to Q \models (P \land R) \to Q$$
 * Vacuous conditional: $$\neg P \models P \to Q $$
 * Transitivity: $$(P \to Q) \land (Q \to R) \models P \to R$$
 * Simplification of disjunctive antecedents: $$(P \lor Q) \to R \models (P \to R) \land (Q \to R)$$

Tautologies involving material implication include:
 * Reflexivity: $$\models P \to P$$
 * Totality: $$\models (P \to Q) \lor (Q \to P)$$
 * Conditional excluded middle: $$\models (P \to Q) \lor (P \to \neg Q)$$

Discrepancies with natural language
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. 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.

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. Recent work in formal semantics and philosophy of language has generally eschewed material implication as an analysis for natural-language conditionals. In particular, such work has often rejected the assumption that natural-language conditionals are truth functional in the sense that the truth value of "If P, then Q" is determined solely by the truth values of P and Q. Thus semantic analyses of conditionals typically propose alternative interpretations built on foundations such as modal logic, relevance logic, probability theory, and causal models.

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.

Conditionals

 * Corresponding conditional
 * Counterfactual conditional
 * Indicative conditional
 * Strict conditional