Implies

Implies is a logical connective that relates two propositions, P and Q, by expressing that Q follows from P. In symbolic notation it is written as P → Q. In classical propositional logic, the connective is defined truth-functionally: the implication is false only when P is true and Q is false; in all other cases it is true. Equivalently, P → Q is logically equivalent to ¬P ∨ Q.

This interpretation is known as material implication. It is standard in mathematics and computer science, but can mislead in natural language where "if" often signals causation, relevance, or obligation rather than mere truth-functional dependence. Consequently, an everyday reading of "If P, then Q" may diverge from the formal truth conditions of P → Q.

Examples help illustrate: P = "x is even" and Q = "x^2 is even." Then P → Q is true for all integers x, since every even number squared is even. Another common case: P = "It is raining" and Q = "The streets are wet." In ordinary language this might reflect a causal expectation, but in logic the truth of P → Q depends only on the truth values of P and Q, not on the weather mechanism.

Related concepts include contrapositive (P → Q is equivalent to ¬Q → ¬P) and entailment (logical consequence), which are meta-logical relations rather than object-level truth functions. In some logics, stronger notions such as strict implication or causal implication are distinguished from material implication.