<logic> A statement of the form, "if A, then B," when A and B stand for wffs or propositions. The wff in the if-clause is called the antecedent (also the implicans and protasis). The wff in the then-clause is called the consequent (also the implicate and apodosis). As a truth function, see material implication. Also called a conditional, or a conditional statement.
Logical implication
A tautologous statement of material implication (next)
Material implication
A truth function that is false when its antecedent is true and its consequent false, and true otherwise. Also the connective that denotes this function; also the compound proposition built from this connective. Notation: p => q (or a thin right arrow). A => B is true unless A is true and B is false. The truth table is
A B | A -> B ----+------- F F | T F T | T T F | F T T | TIt is surprising at first that A => B is always true if A is false, but if X => Y then we would expect that (X & Z) => Y for any Z. This truth function is rarely what implication or "if...then" means in English, but it captures the logical core of that usage and is truth-functional.
Paradoxes of material implication
Two consequences of the formal definition of material implication that violate informal intuitions about implication: (1) that a material implication is true whenever its antecedent is false, and (2) that a material implication is true whenever its consequent is true. These so-called paradoxes do not create contradictions.
[Glossary of First-Order Logic] and [FOLDOC]
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