Material conditional

A truth-functional connective, usually written p → q, that is false only when p is true and q is false. In every other case it is true.

Truth table: A table listing a logical expression’s truth value for every assignment of truth values to its variables. Its four rows define the material conditional’s truth conditions.
Proposition: A declarative content that can be evaluated as true or false. The conditional connects propositions treated as truth-valued units.
Strict implication: A modal relation holding when one proposition necessarily implies another. It strengthens material implication by requiring necessity, not mere truth-functional sufficiency.
Modus ponens: An inference rule that derives q from p and p → q. It is a basic valid inference rule for the material conditional.
Logical implication: A relation in which one proposition guarantees or entails another. Material implication is truth-functional, while logical implication concerns entailment across interpretations.
Truth value: The status of a proposition as true or false in a given interpretation. The connective’s output depends only on its components’ truth values.
Relevant logic: A family of nonclassical logics requiring a meaningful connection between premises and conclusions. Its systems reject some classical conditional inferences that permit irrelevant antecedents.
Modus tollens: An inference rule that derives ¬p from p → q and ¬q. The conditional’s truth conditions make this inference valid.
Disjunction: A logical connective that is true when at least one of its component propositions is true. The conditional p → q is equivalent to ¬p ∨ q in classical logic.
Truth function: A function that assigns an output truth value solely from its input truth values. The material conditional is one particular truth function of two inputs.

Linked from 21 pages