Inductive reasoning is inference in which premises provide support for a conclusion without logically guaranteeing its truth. It extends what is known or assumed to cases, patterns, or explanations not established by the premises alone. Unlike valid deductive reasoning, induction can lead from true premises to a false conclusion. It is central to epistemology, philosophy of science, and the study of how evidence supports beliefs. Although often described as reasoning from particular observations to general rules, it also includes predictions about individual unobserved cases. (plato.stanford.edu)
Logical character and related forms
The defining distinction between induction and deduction concerns support rather than the direction from “particular” to “general.” Logical validity requires that no logically possible situation make all an argument’s premises true and its conclusion false. An inductive argument instead makes its conclusion more credible to some degree, while leaving alternatives possible. Its conclusion is therefore revisable when additional evidence changes the assessment. (plato.stanford.edu)
Induction is commonly called ampliative: its conclusion goes beyond the information logically entailed by its premises. Abductive reasoning, or inference to the best explanation, is also ampliative. In classifications that distinguish them, induction projects observed patterns, whereas abduction selects a hypothesis because of its explanatory merits. Broader uses of “induction” encompass both, so terminology varies across philosophical traditions. (plato.stanford.edu)
Despite its name, mathematical induction is a deductive method. A base case and an appropriate inductive step establish a proposition for every member of the relevant sequence or structure; examining many successful examples does not provide the same kind of proof. (plato.stanford.edu)
Common patterns
Enumerative induction moves from observed instances to a broader claim. For example, repeated observations of objects with properties A and B may support the conclusion that all, or most, A objects have property B. A universal conclusion is stronger than a claim about a proportion and remains vulnerable to a genuine counterexample. Predictive induction instead projects the pattern onto an unobserved instance, such as expecting the next A object to have property B. Neither conclusion follows necessarily from a finite collection of observations. (plato.stanford.edu)
Statistical generalization estimates population characteristics from a sample. Here, inductive strength depends on how the observations were selected, not merely their number. Statistics distinguishes sampling variation from systematic selection bias. A large sample collected through a biased procedure may misrepresent its target population, whereas randomized sampling permits uncertainty to be analyzed under specified assumptions. Representativeness is not guaranteed by size alone. (openstax.org)
Historical development
In ancient Greece, Aristotle discussed epagōgē, usually translated as induction, as movement from particulars toward universals. In the Posterior Analytics, a process involving perception, memory, and experience helps explain the acquisition of first principles from which scientific demonstrations proceed. This account should not simply be equated with modern statistical inference. (plato.stanford.edu)
Francis Bacon developed an influential program of investigative induction in the Novum Organum, published in 1620. He criticized simple accumulation of favorable instances and proposed systematic investigation using experiments, comparisons, and exclusions. His method sought to identify the underlying forms of natural phenomena rather than merely record recurring associations. (plato.stanford.edu)
David Hume gave the classic formulation of the problem of induction in A Treatise of Human Nature (1739). The problem concerns the justification for extending observed regularities beyond the cases already experienced. It became a central challenge for accounts of empirical knowledge and scientific reasoning. (plato.stanford.edu)
Justification and projectibility
In a standard reconstruction of Hume’s challenge, a deductive argument cannot establish that unobserved cases resemble observed ones: a change in regularity is logically conceivable. Appealing to induction’s previous success appears circular, because that appeal itself projects a past pattern into an unobserved case. Hume explained the formation of such expectations through habit or custom rather than a demonstrative justification. (plato.stanford.edu)
Nelson Goodman’s “new riddle of induction” concerns which predicates can legitimately be projected. His invented predicate grue applies to objects observed before a specified time if they are green, and to other objects if they are blue. Observations of green emeralds before that time support both “green” and “grue” descriptions, yet their projections yield incompatible expectations. Agreement with existing observations alone therefore does not determine a unique generalization. (plato.stanford.edu)
Karl Popper accepted the force of Hume’s criticism and proposed an account of science centered on conjectures and attempted falsification, rather than inductive verification. He emphasized the logical asymmetry between confirming instances and counterinstances to universal claims. This is a philosophical position about scientific method, not an uncontested description of all scientific practice. (plato.stanford.edu)
Probability and computational learning
Probabilistic accounts represent evidential support using probability. In Bayesian inference, Bayes’ theorem relates a hypothesis’s prior probability, the likelihood of the evidence under that hypothesis, and its posterior probability. These frameworks distinguish evidence that raises a hypothesis’s probability from evidence that makes it highly probable overall. Their assessments depend on the hypotheses and probability assignments used; probabilistic updating does not itself settle every philosophical question about induction. (plato.stanford.edu)
In machine learning, inductive learning constructs generalizations from training data to handle new instances. Multiple generalizations can agree with the same observations while disagreeing elsewhere. An inductive bias supplies a basis for selecting among them, such as restrictions on expressible hypotheses or preferences built into the search procedure. In this technical sense, bias is not necessarily prejudice or error: it identifies assumptions enabling conclusions beyond the observed examples. (cs.cmu.edu)