Logical implication

Logical implication is a relation between statements that is false only when its premise is true and its conclusion false. In classical logic, it is represented by the material conditional.

Material conditional: A truth-functional connective that is false only when its antecedent is true and its consequent false. It gives the truth-functional reading of implication stated in the definition.
Proposition: A declarative statement that can be assigned a truth value. Implication relates propositions rather than terms or questions.
Strict implication: A modal relation in which one proposition necessarily implies another. Unlike material implication, it requires necessity rather than just a truth-functional condition.
Propositional calculus: A formal system for reasoning about propositions and their logical connectives. It treats implication as a basic connective with formal inference rules.
Truth table: A table listing a compound proposition's truth value for every assignment of truth values to its components. Its four rows make the implication's sole false case explicit.
Antecedent: The proposition that occupies the premise position in a conditional. Its truth is tested against the consequent to determine when the conditional is false.
Entailment: A relation in which truth of premises guarantees truth of a conclusion under specified interpretations. It is often distinguished from the truth-functional connective used to represent implication.
Predicate logic: A logical system that represents objects, properties, relations, and quantification. Implication combines quantified statements in arguments about objects and relations.
Modus ponens: An inference rule that derives Q from P and P implies Q. It shows how an implication licenses a familiar deductive step.
Consequent: The proposition that occupies the conclusion position in a conditional. A false consequent makes the implication false when its antecedent is true.

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