inductive

Inductive is an adjective used across disciplines to describe methods that derive general conclusions, rules, or structures from particular instances or constructions. In logic, inductive reasoning moves from observed cases to probable generalizations, with conclusions that are not guaranteed to be true but are supported by evidence. For example, seeing many white swans may lead to the belief that all swans are white, a generalization that can be falsified by counterexamples.

In mathematics, induction most often refers to a proof technique rather than the broader everyday sense. Mathematical induction proves a statement for all natural numbers by establishing a base case and an inductive step: if the statement holds for one number, it holds for the next; by chaining these steps, it holds universally. A stronger variant, strong induction, assumes the statement holds for all smaller cases to prove the next one. Inductive concepts also appear in defining objects, such as inductive definitions that specify how to construct members of a set.

Inductive definitions are common in computer science and logic. They define data structures or sets by specifying constructors, and proofs about these structures often use structural induction, a form of induction on the structure’s construction.

In computer science and artificial intelligence, inductive learning or inference refers to deriving general rules from specific examples (training data). Algorithms build models that generalize to new data, guided by an inductive bias—the assumptions that shape learning. Inductive approaches contrast with deductive methods, which derive specific conclusions from general rules, and with purely empirical methods, which may rely on experimentation without formal generalization guarantees.

In engineering and physics, induction also denotes electromagnetic induction, the generation of electric current by changing magnetic flux, a foundational principle behind transformers and inductors.