implication

Implication is a relation between two propositions that expresses that one proposition follows from another. In formal logic, it is written as P → Q, read as “if P then Q.” The antecedent P is the condition, and the consequent Q is the result or consequence.

The standard truth-functional reading of material implication defines the truth of P → Q by the truth values of P and Q: it is false only when P is true and Q is false; in all other cases it is true. This makes P → Q logically equivalent to ¬P ∨ Q (not P or Q). The concept is central to formal deduction and proof construction, where the truth of Q is guaranteed by the truth of P under the implication.

Key related notions include the contrapositive and the converse. The contrapositive of P → Q is ¬Q → ¬P, which is logically equivalent to P → Q. The converse, Q → P, is not generally equivalent to P → Q and can have a different truth value.

Beyond formal logic, implication also refers to what a statement suggests or indicates in ordinary language. An implication may be intended or inferred without being explicitly stated, and it can be ambiguous or context-dependent. In philosophy and modal logic, stronger concepts such as strict or necessary implication are used, distinguishing merely material implication from relations that hold under possible interpretations or in all relevant models.

Applications of implication span mathematics, computer science, and everyday reasoning, where it underpins conditional statements, proofs, algorithms, and inferential thinking.