The material conditional (also known as material implication) is a binary operation commonly used in logic. When the conditional symbol → {\displaystyle \to } is interpreted as material implication, a formula P → Q {\displaystyle P\to Q} is true unless P {\displaystyle P} is true and Q {\displaystyle Q} is false. 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 → {\displaystyle \to } (U+2192 → RIGHTWARDS ARROW). The material conditional is also notated using the infixes ⊃ {\displaystyle \supset } and ⇒ {\displaystyle \Rightarrow } (U+2283 ⊃ SUPERSET OF and U+21D2 ⇒ RIGHTWARDS DOUBLE ARROW respectively). In the prefixed Polish notation, conditionals are notated as C p q {\displaystyle Cpq} . In a conditional formula p → q {\displaystyle p\to q} , the subformula p {\displaystyle p} is referred to as the antecedent and q {\displaystyle 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 → q ) → ( r → s ) {\displaystyle (p\to q)\to (r\to s)} .
History In Arithmetices Principia: Nova Methodo Exposita (1889), Peano expressed the proposition "If A {\displaystyle A} , then B {\displaystyle B} " as A {\displaystyle A} Ɔ B {\displaystyle B} with the symbol Ɔ, which is the opposite of C. He also expressed the proposition A ⊃ B {\displaystyle A\supset B} as A {\displaystyle A} Ɔ B {\displaystyle B} . Hilbert expressed the proposition "If A, then B" as A → B {\displaystyle 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 ⊃ B {\displaystyle A\supset B} . Following Russell, Gentzen expressed the proposition "If A, then B" as A ⊃ B {\displaystyle A\supset B} . Heyting expressed the proposition "If A, then B" as A ⊃ B {\displaystyle A\supset B} at first but later came to express it as A → B {\displaystyle A\to B} with a right-pointing arrow. Bourbaki expressed the proposition "If A, then B" as A ⇒ B {\displaystyle A\Rightarrow B} in 1954.
Semantics
Truth table 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 the following truth table:
One can also consider the equivalence A → B ≡ ¬ ( A ∧ ¬ B ) ≡ ¬ A ∨ B {\displaystyle A\to B\equiv \neg (A\land \neg B)\equiv \neg A\lor B} . The conditionals ( A → B ) {\displaystyle (A\to B)} where the antecedent A {\displaystyle A} is false, are called "vacuous truths". Examples are ...
... with B {\displaystyle B} false: "If Marie Curie is a sister of Galileo Galilei, then Galileo Galilei is a brother of Marie Curie." ... with B {\displaystyle B} true: "If Marie Curie is a sister of Galileo Galilei, then Marie Curie has a sibling."
Analytic tableaux
Formulas over the set of connectives { → , ⊥ } {\displaystyle \{\to ,\bot \}} are called f-implicational. In classical logic the other connectives, such as ¬ {\displaystyle \neg } (negation), ∧ {\displaystyle \land } (conjunction), ∨ {\displaystyle \lor } (disjunction) and ↔ {\displaystyle \leftrightarrow } (equivalence), can be defined in terms of → {\displaystyle \to } and ⊥ {\displaystyle \bot } (falsity):
… excerpt ends here. Continue reading the full article.
