The term is most widely used for any process of reasoning that takes us from empirical premises to empirical conclusions supported by the premises, but not deductively entailed by them. Inductive arguments are therefore kinds of ampliative argument, in which something beyond the content of the premises is inferred as probable or supported by them. Induction is, however, commonly distinguished from arguments to theoretical explanations, which share this ampliative character, by being confined to inference in which the conclusion involves the same properties or relations as the premises. The central example is induction by simple enumeration, where from premises telling that Fa, Fb, Fc…, where a, b, c, are all of some kind G, it is inferred that Gs from outside the sample, such as future Gs, will be F, or perhaps that all Gs are F. If this, that, and the other person deceive them, children may well infer that everyone is a deceiver. Different but similar inferences are those from the past possession of a property by some object to the same object’s future possession of the same property, or from the constancy of some law-like pattern in events and states of affairs to its future constancy: all objects we know of attract each other with a force inversely proportional to the square of the distance between them, so perhaps they all do so, and always will do so.

The rational basis of any such inference was challenged by Hume, who believed that induction presupposed belief in the uniformity of nature, but that this belief had no defence in reason, and merely reflected a habit or custom of the mind. Hume was not therefore sceptical about the propriety of processes of induction, but sceptical about the role of reason in either explaining it or justifying it. Trying to answer Hume and to show that there is something rationally compelling about the inference is referred to as the problem of induction. It is widely recognized that any rational defence of induction will have to partition well-behaved properties for which the inference is plausible (often called projectible properties) from badly behaved ones for which it is not (see Goodman’s paradox). It is also recognized that actual inductive habits are more complex than those of simple enumeration, and that both common sense and science pay attention to such factors as variations within the sample giving us the evidence, the application of ancillary beliefs about the order of nature, and so on (see Mill’s methods). Nevertheless, the fundamental problem remains that any experience shows us only events occurring within a very restricted part of the vast spatial and temporal order about which we then come to believe things. See also confirmation, explanation, falsification, vindication.