The Problem of Induction

1. Scope and evidence

The core reading of Hume's argument (the demonstrative/probable dilemma and the circularity of any empirical justification of the Uniformity Principle) is Established and is presented in essentially this form by the Stanford Encyclopedia of Philosophy, the Internet Encyclopedia of Philosophy and standard scholarship. The precise interpretation of Hume's terms ("demonstrative", "probable") and of his ultimate intent (radical scepticism versus a narrower attack on rationalism, or a thesis in cognitive psychology) is Disputed among specialists. The claim that the problem remains unsolved is Shared/Traditional — the mainstream view — though a minority of philosophers argue for partial or full solutions. We flag explicitly: no proposed solution commands consensus, and the problem is a live, open issue in philosophy of science today.

2. From causation and custom to induction (recap of Card #5)

As Card #5 set out, Hume's empiricism traces all ideas back to sensory impressions (the Copy Principle), and his analysis of causation concludes that we never perceive a necessary connection between cause and effect — only their "constant conjunction". What carries us from an observed cause to an expected effect is not reason but custom or habit, a propensity of the imagination. The problem of induction is the epistemological edge of that same analysis. Card #5 deliberately deferred it to here: having shown that causal inference rests on experience of constant conjunction, Hume now asks what justifies extending that experience to cases we have not observed — above all, to the future. His answer is that nothing in reason justifies it.

For Hume, causation is the only relation "which can lead us beyond the immediate impressions of our memory and senses". When we infer that the next piece of bread will nourish us, or that the sun will rise tomorrow, we rely on a transition of the form: "I have found that such an object has always been attended with such an effect, and I foresee, that other objects, which are, in appearance, similar, will be attended with similar effects" (Enquiry 4.16). Hume's question is not whether we make such inferences — obviously we do, and must — but what their "foundation" is: the "logic" or "process of argument" that supports them.

3. Hume's argument in full: the demonstrative/probable dilemma

Hume identifies a hidden assumption behind every inductive inference. In the Treatise he puts it thus: if reason determined us, "it wou'd proceed upon that principle, that instances, of which we have had no experience, must resemble those, of which we have had experience, and that the course of nature continues always uniformly the same" (T 1.3.6.4 / SBN 89). Commentators call this the Uniformity Principle (also the Resemblance Principle or Principle of the Uniformity of Nature).

Hume then argues, by a dilemma, that this principle can have no rational support. He divides all reasoning into two kinds: demonstrative reasoning (concerning "relations of ideas" — mathematics, logic, anything "intuitively or demonstratively certain") and probable or "moral" reasoning (concerning "matters of fact and existence"). This division is often called Hume's fork.

First horn — no demonstrative argument. A demonstrative argument establishes a conclusion whose denial is a contradiction. But there is no contradiction in supposing the course of nature to change: "it implies no contradiction that the course of nature may change, and that an object seemingly like those which we have experienced, may be attended with different or contrary effects" (Enquiry 4.18). We can clearly and distinctly conceive of bread that looks identical yet poisons us, or of a sunless tomorrow. Since the Uniformity Principle can be denied without contradiction, no demonstrative/a priori argument can prove it. Hume applies this directly to the bread example: "The bread, which I formerly eat, nourished me; that is, a body of such sensible qualities, was, at that time, endued with such secret powers: But does it follow, that other bread must also nourish me at another time, and that like sensible qualities must always be attended with like secret powers? The consequence seems nowise necessary" (E 4.16 / SBN 33–34).

Second horn — no probable argument (circularity). Could experience justify the principle? No — because all reasoning from experience already presupposes it. As Hume writes in the Enquiry: "all our experimental conclusions proceed upon the supposition, that the future will be conformable to the past. To endeavour, therefore, the proof of this last supposition by probable arguments … must be evidently going in a circle, and taking that for granted, which is the very point in question" (E 4.19 / SBN 35–36). And, famously: "It is impossible, therefore, that any arguments from experience can prove this resemblance of the past to the future; since all these arguments are founded on the supposition of that resemblance" (E 4.21 / SBN 37–38). To argue "uniformity has held so far, therefore it will continue to hold" is to use an inductive inference to license induction — assuming precisely what is in question. (Note: the "conformable to the past" line at E 4.19 and the "resemblance of the past to the future" line at E 4.21 are two distinct sentences roughly two paragraphs apart, not a single continuous quotation.)

Both horns fail, so there is no chain of reasoning from the premises of an inductive inference to its conclusion. Hume's conclusion in the Treatise: "even after experience has inform'd us of their constant conjunction, 'tis impossible for us to satisfy ourselves by our reason, why we shou'd extend that experience beyond those particular instances, which have fallen under our observation" (T 1.3.6.11).

The Stanford Encyclopedia reconstructs the argument as a chain of premises: (P1) there are only demonstrative and probable arguments; (P2) inference presupposes the Uniformity Principle; the first horn yields "no demonstrative argument for UP", the second "no probable argument for UP"; therefore "no argument for UP"; therefore the inductive inference is not justified. It is worth stressing that Hume's target is specifically enumerative induction ("all observed A have been B; the next A will be B"), though the point generalises.

A crucial nuance the site flags for accuracy. Hume did not conclude that we should stop making inductive inferences, nor did he think them worthless. He is not a practical sceptic. His point is that their foundation is not reason. His own "sceptical solution" (Enquiry §V) is psychological: the inference is produced by custom or habit, "a species of natural instincts, which no reasoning or process of the thought and understanding is able, either to produce, or to prevent". Nature has not left something so important to our fallible reasoning; the inductive propensity is a kind of "pre-established harmony" between the course of nature and the succession of our ideas. Scholars therefore debate whether Hume is a radical inductive sceptic, a "naturalist" describing how the mind works, or (per Don Garrett and David Owen) advancing a thesis in cognitive psychology about the faculty of reason rather than a normative verdict.

4. The Uniformity of Nature principle

The Uniformity Principle is the load-bearing assumption of Hume's argument. It is not a premise of any particular inductive inference stated explicitly; rather, it is the tacit supposition that would be needed to turn the inference into a valid piece of reasoning. Hume's dilemma is really an attack on this principle: it can be justified neither a priori (a non-uniform nature is conceivable) nor empirically (that would be circular). Some later reconstructions dispense with an explicit Uniformity Principle and press the circularity point directly against any attempt to argue from observed to unobserved; but the circularity worry survives either way. It is also worth noting that "uniformity of nature" is slippery — nature is manifestly not uniform in every respect (weather changes, species go extinct), so a defensible version must specify in what respects and which regularities are projected — which is exactly the doorway to Goodman's later riddle.

5. Goodman's New Riddle of Induction (grue) — and how it differs from Hume

Nelson Goodman, in Fact, Fiction, and Forecast (1955), granted that Hume had essentially identified something real, but argued that the old problem of justifying induction should give way to a sharper, "new" riddle about confirmation. Goodman's target was the syntactic confirmation theories of Carnap and Hempel, which assumed that a generalisation is confirmed by its instances.

Goodman defined the predicate "grue" relative to an arbitrary future time t: an object is grue if it is examined before t and found green, or not so examined and is blue. (He similarly defined "bleen": blue if examined before t, otherwise green.) Now consider emeralds. Every emerald examined so far has been green — and, by construction, every one has equally been grue (since all were examined before t and were green). The very same evidence supports two generalisations:

"All emeralds are green" → the next (post-t) emerald will be green;

"All emeralds are grue" → the next (post-t, unexamined) emerald will be blue.

The evidence is identical; the predictions are incompatible. Inductive logic, considered purely formally, gives no reason to prefer one. (Wikipedia's standard illustration fixes t at 1 January 2036: before that date green things and grue things are indistinguishable; only from 2 January 2036 do previously green, now-unexamined emeralds count as "bleen" and the predictions diverge.) Goodman's moral: confirmation cannot depend on logical form (syntax) alone; there can be no purely formal logic of induction. The real task is to distinguish projectible predicates (like "green"), which are legitimately extended to unobserved cases, from non-projectible ones (like "grue"). Goodman's own proposed criterion was entrenchment — projectible predicates are those with a track record of successful past projection in our linguistic practice.

The distinction the site insists on. Hume's problem asks: what justifies projecting past regularities into the future at all? Goodman's riddle asks: given that we do project, which regularities/predicates are the right ones to project? The new riddle bites even if one grants that induction from past to future is legitimate in principle; it is therefore a distinct problem, not a restatement of Hume's. It is a mistake — common in popular accounts — to treat "grue" as merely a colourful retelling of Hume. (A further subtlety, noted by Hilary Putnam and others: the time-dependence of "grue" is language-relative — one could take grue and bleen as primitive and define "green" disjunctively in their terms — which is part of why the riddle is so stubborn.)

6. Major proposed responses — surveyed, none decisive

(a) Popper's falsificationism. Karl Popper accepted Hume's negative conclusion — induction cannot be justified — and argued that science does not actually use induction. Instead, science advances by conjecture and refutation: scientists propose bold, falsifiable hypotheses, deduce testable predictions, and try to refute them; a theory that survives severe testing is "corroborated" but never verified or shown probable. The strength of the view is that it respects the logical asymmetry between verification (impossible for universal laws) and falsification. But the mainstream assessment is that it does not so much solve Hume's problem as avoid the question — and possibly relocate it. Critics (Wesley Salmon, Richard Jeffrey, Colin Howson) note that (i) if we must choose which corroborated theory to act on or believe for future predictions, we seem to need an inductive step after all; (ii) falsification is not clean, because observation statements are themselves fallible and no hypothesis is conclusively refuted (the Duhem–Quine point); and (iii) as a description of scientific practice it fits poorly (Kuhn showed scientists do not abandon paradigms at the first anomaly). John Norton characterises falsificationism as having "risen and fallen". So: an influential re-framing, not a consensus solution.

(b) Bayesian / probabilistic approaches. The dominant modern framework treats inductive support not as certainty but as degree of belief updated by evidence via Bayes's theorem: a hypothesis's probability rises as confirming evidence accumulates. This captures much scientific and statistical reasoning and dissolves the demand for certainty that made Hume's problem look hopeless. But whether it solves Hume is contested. Colin Howson (Hume's Problem, 2000) argues Bayesianism supplies a "logic" of consistent inductive inference — yet concedes that inductive conclusions still require an empirical premise (the prior), so the justification is conditional, not absolute: "Inductive reasoning is justified to the extent that it is sound, given appropriate premises." Critics reply that the choice of prior probabilities is exactly where Hume's circularity reappears — nothing in the probability calculus tells you which prior to adopt, and different priors yield different predictions from the same data. Samir Okasha argues Bayesianism does not solve Hume's problem in the classical sense, though he thinks the sceptical challenge cannot be transposed unchanged into a degree-of-belief setting. A recent verdict (Chloé de Canson, Philosophers' Imprint, 2025) is that the Bayesian "inferential solution" fails by its own lights because conditionalisation cannot be justified independently of the priors. Consensus: a powerful reformulation, widely held not to defeat the underlying problem.

(c) Reichenbach's pragmatic vindication. Hans Reichenbach conceded that induction cannot be shown to be reliable (that would beg Hume's question), but argued it can be pragmatically vindicated: if any method of prediction will work — that is, if nature has any limiting frequency to discover — then the inductive "straight rule" is guaranteed to find it in the long run. So induction is our best bet: we lose nothing by using it and may gain everything. The appeal is that it asks for no unprovable premise about nature's uniformity. The much-discussed weakness is that the argument does not single out induction uniquely: infinitely many alternative "asymptotic" rules also converge in the limit, and the vindication says nothing about the short run, where we actually live. Wesley Salmon spent years trying, without agreed success, to add constraints (simplicity, normalising conditions) that would privilege the straight rule. A serious idea, generally judged incomplete.

(d) Strawson's ordinary-language dissolution. P.F. Strawson (Introduction to Logical Theory, 1952) argued the problem rests on a confusion. To ask for a deductive justification of induction is to demand that induction "be shown to be really a kind of deduction" — an incoherent demand. To ask whether induction in general is justified/reasonable is, Strawson says, like asking whether the law is legal: being reasonable about matters of empirical fact just means proportioning belief to inductive evidence. So the demand for an external justification of induction as a whole is a kind of category mistake; the genuine questions concern whether particular inductive inferences meet our inductive standards. Assessment: many find this deflationary move too quick — it arguably answers a question about our concept of reasonableness while leaving Hume's worry (why is conforming to those standards a reliable guide to truth?) untouched. Influential, but widely thought to dissolve the wording rather than the problem.

(e) Kant's transcendental reply (brief — see Card #5). As noted on Card #5 regarding causation, Immanuel Kant took Hume to have wakened him from "dogmatic slumber" and responded that causation — and, on some readings (De Pierris & Friedman), the uniformity of nature — is a synthetic a priori condition of the very possibility of experience: the mind necessarily structures experience under the category of cause. If so, uniformity is not an empirical hypothesis we must justify but a precondition of there being ordered experience at all. Critics reply that this secures at most that we must think in causal/uniform terms, not that nature is uniform in the specific ways our inductions require — and it purchases the result at the high price of transcendental idealism. Relevant to induction, but not a settled solution.

Other contemporary work. Formal learning theory and Gerhard Schurz's meta-induction programme (Hume's Problem Solved: The Optimality of Meta-Induction, MIT Press, 2019) argue that, although we cannot justify induction's reliability non-circularly, we can prove a priori that a meta-inductive strategy is optimal. Specifically, Schurz demonstrates that "a regret-based learning strategy, attractivity-weighted meta-induction, is predictively optimal in all possible worlds among all prediction methods accessible to the epistemic agent"; the a priori justification of meta-induction is then said to generate a non-circular a posteriori justification of ordinary "object" induction. The result is mathematically serious but its philosophical sufficiency is debated — Tomoji Shogenji (Journal for General Philosophy of Science, 2023) identifies a "twofold tension" in the proposal — and the title's claim is not generally accepted as closing the question.

7. Current status of the problem

There is no consensus that Hume's problem has been solved. The Stanford Encyclopedia's survey ends with a section on "living with inductive skepticism"; C.D. Broad's epigram — from his 1926 Bacon tercentenary address, where he looked forward to a day when "Inductive Reasoning, which has long been the glory of Science, will have ceased to be the scandal of Philosophy" — is still routinely quoted to describe the standoff. Bertrand Russell went so far as to write, in A History of Western Philosophy (1946, p. 699), that if Hume's problem cannot be solved "there is no intellectual difference between sanity and insanity." In practice, working scientists and statisticians do not wait on a solution: they use inductive, hypothetico-deductive and (increasingly) explicitly Bayesian methods, judging them by their track record. This is much the pragmatic attitude A.J. Ayer recommended in Language, Truth and Logic: "It is time, therefore, to abandon the superstition that natural science cannot be regarded as logically respectable until philosophers have solved the problem of induction." Philosophers of science largely agree that Hume identified something genuine and deep; they disagree about whether it demands a solution, a dissolution, or simply honest acknowledgement. The problem thus remains open — arguably the clearest example of a simply-stated question that has resisted resolution for over two and a half centuries. (Card #8 will examine one distinctively Scottish response: Thomas Reid's Common Sense philosophy, which meets Humean scepticism not by defeating the argument but by trusting the natural, instinctive beliefs — including the "inductive principle" — that Reid takes to be built into the human constitution.)