Inductive Inference

Inductive inference is the process of deriving general conclusions from particular observations — the logical move from 'some swans are white' to 'all swans are white,' or from finite data to a universal law. It is the engine of empirical science, everyday prediction, and learning theory, yet it has resisted complete formalization since Hume first identified the problem of induction in 1748. No finite set of observations logically entails a generalization; inductive inference is always ampliative, going beyond the evidence it rests upon.

The modern treatment of inductive inference divides into three streams. The logical tradition, from Carnap to Solomonoff, attempts to define a measure of confirmation or a universal prior that rationalizes the leap from evidence to hypothesis. The computational tradition, rooted in computational learning theory, replaces logical entailment with resource-bounded learnability: what can be inferred given bounded time, data, and memory? The cognitive tradition, drawing on psychology and neuroscience, asks how biological agents perform inductive inference with heuristics that are demonstrably non-optimal yet remarkably effective.

The unifying insight across these streams is that induction is not a deficiency of finite minds but a property of any system embedded in a structured world. Where the structure of the world matches the structure of the inferencer, induction succeeds. The mismatch — between the compressibility of reality and the representational capacity of the learner — is where no-free-lunch limits bite and where generalization fails.