induction (logic)

induction in logic , a form of argument in which the premises give grounds for the conclusion but do not necessitate it. Induction is contrasted with deduction , in which true premises do necessitate the conclusion. An important form of induction is the process of reasoning from the particular to the general. Francis Bacon in his Novum Organum (1620) elucidated the first formal theory of inductive logic, which he proposed as a logic of scientific discovery, as opposed to deductive logic, the logic of argumentation. Both processes, however, are used constantly in research. By observation of events (induction) and from principles already known (deduction), new hypotheses are formulated; the hypotheses are tested by applications; as the results of the tests satisfy the conditions of the hypotheses, laws are arrived at—by induction; from these laws future results may be determined by deduction. David Hume has influenced 20th-century philosophers of science who have focused on the question of how to assess the strength of different kinds of inductive argument (see Nelson Goodman ; Sir Karl Raimund Popper ). For a classic account of inductive arguments see J. S. Mill, System of Logic (1843).

Bibliography: See also R. Swinburne, ed., The Justification of Induction (1974); J. Cohen, An Introduction to the Philosophy of Induction and Probability (1989).

induction
1. A method of logical inference in which a general but not necessarily true conclusion is drawn from a set of particular instances. In machine learning, for example, the term induction is used to describe an approach to machine learning in which generalized structures or statements are inferred from particular examples.

2. A process for proving mathematical statements involving members of an ordered set (possibly infinite). There are various formulations of the principle of induction. For example, by the principle of finite induction, to prove a statement P(i) is true for all integers i ≥ i₀, it suffices to prove that

(a) P(i₀) is true;

(b) for all k ≥ i₀, the assumption that P(k) is true (the induction hypothesis) implies the truth of P(k+1).

(a) is called the basis of the proof, (b) is the induction step.

Generalizations are possible. Other forms of induction permit the induction step to assume the truth of P(k) and also that of P(k–1), P(k–2), …, P(k–i)

for suitable i. Statements of several variables can also be considered. See also structural induction.