Falsificationism gives an attractive account of one aspect of scientific progress: replace protected dogma by conjectures that make risky claims and expose them to serious tests. From $T\wedge A\Rightarrow E$ and an accepted $\neg E$, deduction rules out the conjunction. A theory that accommodates every possible outcome supplies less opportunity to learn from error. Karl Popper emphasizes exposure to refutation and warns against treating any convenient supporting instance as a demanding test.^[ref]

This captures why a successful novel prediction can be more informative than retrospectively adjusting a model to its original fitting data. It also helps articulate a progressive replacement: the new account should address the earlier successes and the discrepancy, while exposing itself to further independent tests. Merely adding a freely adjustable parameter after each failure need not improve knowledge.

The Duhem problem limits the simple account. A failed prediction is a failure of a package containing initial conditions, instrument assumptions and background theory. It is not a deductive identification of the particular target theory as false. Revising an auxiliary hypothesis can be rational when independently supported, but an unconstrained rescue can frustrate testing. Methodological judgement is needed to distinguish them; the bare logic of falsifiability does not do all the work.

There is also a positive-warrant problem. A theory's survival of past tests does not deductively entail that it will succeed tomorrow. Calling it corroborated records its demanding test history, but does not by itself solve the problem of induction. Nor does the elimination of one theory guarantee that a chosen rival is closer to the truth: many alternatives may remain, and progress can involve improved measurement, new concepts or increased explanatory scope as well as refutation.

A convincing form of falsificationism is therefore a critical methodology within a broader account. It explains the value of error detection, risky commitments and resistance to immunizing manoeuvres. It becomes less convincing as a complete logical algorithm for identifying falsified theories, ranking every successor or deriving confidence about future performance. A more adequate account combines severe criticism with independent checks on assumptions, comparative success and explicit acknowledgement of fallible inference.

Incommensurability can express several different claims. Rival scientific paradigms may organize terms differently, favour different exemplary problems, or rank standards of explanation differently. None of those claims by itself entails that their users cannot communicate or that no evidence can favour one account. Distinguishing these levels is essential to judging the claim rather than treating it as a slogan.

Semantic change is a serious possibility. In special relativity, distant simultaneity depends on an inertial frame; a Newtonian framework instead admits an absolute temporal ordering. Reusing the word “simultaneous” does not preserve every inference across the two accounts. A translation must identify which concepts and conditions correspond. That is more demanding than matching terms one by one.

Yet difference does not imply total incomparability. Observers can agree on local clock readings and compare predictions for specified experimental arrangements. In the low-speed domain, an appropriate limit relates many relativistic results to their Newtonian counterparts. Such a connection does not make the theories identical: it identifies a domain and an approximation under which particular predictions agree. It supplies a basis for comparison without requiring a complete, context-free translation of every theoretical term.

Methodological differences can also be real without making theory choice arbitrary. One framework may solve a familiar problem especially well, while another opens a wider range of questions. Accuracy, scope, simplicity and coherence can conflict, and applying them may require judgement. But shared measurements and successful novel predictions can still constrain that judgement. A disagreement over their relative weight is not evidence that every outcome is equally acceptable.

A defensible reading of Kuhn's revolutionary theory change therefore involves partial changes of meaning and standards, together with the work needed to understand a rival framework. It need not involve permanent mutual unintelligibility.^[ref] The strongest thesis, that rival frameworks cannot be rationally compared at all, is much less plausible: it would erase the distinction between changing conceptual resources and losing all evidential contact. The moderate thesis usefully warns against assuming that scientific change is always the addition of sentences to an unchanged language.

The history of science is relevant because a philosophical account purporting to describe scientific knowledge and rational change must fit more than the finished derivations in current textbooks. Historical episodes reveal what evidence was available, which alternatives were considered and how assumptions about instruments and phenomena were revised. A reconstruction that imports later knowledge can make a reasonable contemporary choice appear irrational.

A simple picture of falsificationism illustrates the point. A discrepancy between an observed orbit and a calculation need not immediately discredit Newtonian mechanics. The prediction also uses estimated masses, initial data and assumptions about other bodies. Investigating an additional perturbation may be more informative than rejecting the entire theory. History can show whether such a proposal was independently constrained and successful, rather than merely a device for avoiding any possible refutation. The philosophical distinction between a testable extension and an ad hoc hypothesis therefore benefits from examining actual evidence and practice.

Thomas Kuhn's introduction to The Structure of Scientific Revolutions by Thomas Kuhn challenges the picture obtained from retrospective textbook accumulation. His distinction between normal science and revolutionary theory change directs attention to changing research problems, standards and exemplars.^[ref] Whether one accepts that particular account, the historical challenge is substantive: an account of progress must explain why some earlier achievements remain useful while broader frameworks change.

Historical explanation is not itself a complete justification. Showing that a social setting or personal commitment caused someone to accept a theory does not show that the theory was well supported. Nor does a single episode automatically refute a normative ideal: actual scientists can make mistakes. The context of discovery and context of justification can be distinguished even when they interact. What matters is identifying the standard being evaluated and the evidence relevant to it.

History is therefore neither dispensable decoration nor an infallible rulebook. It can expose counterexamples to oversimple universal claims, supply scientifically realistic cases and reveal unnoticed alternatives. Philosophical reasoning must then assess the inference, the evidential situation and the claimed norm. A good account should be historically responsible without reducing rational justification to whatever scientists happened to do.

A scientist should take a sceptical scenario seriously when it exposes a credible way in which the evidence could be misleading and there is a useful way to investigate that possibility. A shared instrument offset, a software error copied across analyses, or an unrecognized background signal can survive the appearance of repeated agreement. Such possibilities motivate independent calibration, changes of apparatus and analysis, and deliberately varied experiments. Mere repetition using the same error-prone chain is weaker than agreement among genuinely independent tests.

Consider two detectors that report the same departure from a predicted signal. Their agreement supports a new effect less strongly if both use the same calibration source and processing program. A proposed common failure is not defeated merely by observing that both detectors agree. An independently constructed detector, measurements at several settings, or analysis of deliberately hidden test signals can make the failure scenario and the target hypothesis predict different outcomes. Here scientific doubt generates a better test rather than an unlimited demand for reassurance.

A radically global sceptical scenario is different. If it is stipulated to reproduce every possible observation and experiment, it supplies no empirical contrast to investigate. That does not make it logically contradictory or prove it false. It limits its role in choosing among scientific models. A researcher cannot improve a detector by an experiment whose outcome is stipulated to be identical whether all experience is globally deceptive or not.

The line is not simply between imaginable and unimaginable errors. Limited time and resources require attention to plausibility, consequences and discriminating power. A low-probability fault may deserve a check when it could invalidate an entire result and is inexpensive to test. Conversely, inventing an unconstrained auxiliary hypothesis after every contrary result can insulate a favoured theory from criticism. The Duhem problem explains why a failed prediction does not isolate the guilty assumption; it does not make every rescue equally well evidenced.

A scientist can therefore combine fallibilism with justified confidence. Results can be warranted by calibration, independent methods and successful interventions without excluding every logically possible deception. Productive scientific scepticism identifies concrete vulnerabilities and seeks evidence against them. It neither assumes that every appearance is reliable nor treats the absence of Cartesian certainty as the absence of scientific knowledge.

Necessary connections would matter if they provided an independently knowable bridge from observed cases to unobserved ones. Their mere existence would not provide that bridge. In David Hume's problem of induction, observations of past regularities do not deductively entail that the same regularities continue. Appealing to the past reliability of inductive reasoning uses the very kind of inference whose justification is at issue.^[ref]

Suppose, for example, that a particular causal power necessarily makes an isolated system respond in a certain way. If we knew independently that the power exists, that it remains present and that the relevant conditions obtain, a prediction could follow by deduction. But repeated agreement with the prediction does not itself establish those premises without further inductive reasoning. The proposal can relocate the epistemic burden rather than discharge it. A guarantee that every cause has some necessary effect also fails to identify which cause and effect our observations instantiate.

This separates metaphysical necessity from access to it. A scientist might posit causal powers because they unify otherwise separate regularities, explain counterfactual responses and guide successful interventions. An experimentally supported molecular mechanism can make the behaviour of a material intelligible across varied conditions. That supplies a comparative scientific reason to prefer a structured causal model over a bare catalogue. It does not demonstrate that the model captures a connection whose violation is impossible in every relevant world, or that its scope extends beyond the tested domain.

Hume's discussion of necessary connection challenges the idea that inspecting one cause discloses such a connection directly; repeated experience instead explains our expectation.^[ref] A defender of causal powers can reject that account, but must explain both the powers and how we know them. Mathematical necessity in a model is insufficient: the deduction is conditional on the model applying to the physical system.

The defensible conclusion is conditional. Independently warranted knowledge of stable necessary connections could justify particular projections. Simply adding them to one's account of nature does not solve the problem of induction. Scientific support for a causal account can remain strong, comparative and fallible without becoming a noncircular deductive foundation for every future inference.

Instrumentalism treats theoretical descriptions primarily as tools for organizing and predicting observations rather than as a literally true inventory of unobservable entities. Its attraction is epistemic economy: where empirical success is all the evidence establishes, using a successful calculation need not commit us to every mechanism appearing in it. The abandonment of phlogiston illustrates why such caution can be reasonable.

However, distinguish three claims: we can use a theory without believing it; we ought to suspend belief about some of its entities; and those entities do not exist or theoretical sentences are not truth-apt. The first is commonplace and does not establish the last two. Constructive empiricism interprets theoretical claims literally while distinguishing acceptance for empirical adequacy from belief in the whole theory. It therefore offers restraint without treating all theoretical language as a disguised calculation recipe.^[ref]

A scientific realist can answer that independent experimental access, successful interventions and unexpectedly accurate new predictions support some entities more strongly than mere curve fitting. The no-miracles argument treats this convergence as something needing explanation. The reply is not conclusive: explanatory success need not guarantee truth, and selectively retained successful theories can make retrospective success look stronger than it was. Equally, historical failures do not show that every present theoretical commitment is equally fragile.

There is therefore a good argument for instrumental use and selective suspension of belief where evidence does not discriminate ontologies. There is a weaker argument for universal instrumentalism about every scientific entity. A defensible position should compare the actual evidence for particular commitments and explain why empirical adequacy is the right stopping point, rather than infer global agnosticism from the bare possibility of error.

Empirical equivalence must be stronger than agreement with the evidence already collected: relative to a stated domain, empirically equivalent theories yield the same possible observational consequences. If they nevertheless make incompatible claims about unobservable entities, observations in that domain cannot alone determine which ontology is correct. This is a strong form of scientific underdetermination, directed against a unique inference from empirical success to full theoretical truth.

The scientific realist has several responses. One may deny that every formally constructed alternative is a serious physical rival, appeal to independently justified explanatory virtues, or restrict commitment to the structure genuinely shared by the alternatives. If two descriptions are connected by a change of variables or a gauge transformation, apparent difference may be representational rather than a disagreement about the world. Before calling such a pair a counterexample, identify what incompatible physical claims remain.

Actual and merely possible rivals have different evidential force. An actual, comparably developed rival demonstrates that the current inference is nonunique under specified criteria. A merely possible rival refutes a claim of logical entailment if its possibility is established, but does not automatically defeat ordinary fallible justification: otherwise any unexcluded sceptical possibility would defeat almost every empirical belief. A distant conceivable alternative need not deserve the same prior probability or explanatory standing as a well-tested account.

Consequently, empirical equivalence is a serious challenge to unrestricted, uniquely determined theoretical belief. It is not a deductive disproof of all scientific realism. A convincing argument must say whether the target is certainty or defeasible warrant, whether equivalence covers all observations or only present data, and why the rivals are physically distinct. Merely announcing that another story could be invented leaves these crucial questions unanswered.

Let $R(x)$ mean that $x$ is a raven and $B(x)$ that it is black. The hypothesisis logically equivalent to $\forall x[\neg B(x)\Rightarrow \neg R(x)]$. If the observed paper is both nonblack and a nonraven, it is a positive instance of the second formulation. Combining an unrestricted positive-instance rule with preservation of scientific confirmation under logical equivalence produces the raven paradox.^[ref]

Whether the observation actually supplies evidence depends on what was already known and how the object was selected. In Bayesian confirmation, with background $K$, support means $P(H\mid E,K)>P(H\mid K)$. If the observer already knows the selected item is a nonraven and its colour is conditionally independent of $H$, then $P(E\mid H,K)=P(E\mid \neg H,K)$: the likelihood ratio is one and there is no confirmation. If all of $E$ was already known, learning it again also gives no update.

Under a different sampling scheme, the same classification can provide slight support. In the finite-population model of raven confirmation, fix $r$ ravens and $M$ nonblack objects. Compare no nonblack raven with exactly one. Sampling uniformly among the nonblack objects gives a nonraven with probabilities $1$ and $1-1/M$, hence likelihood ratio $(1-1/M{)}^{-1}>1$. If $M$ is large this is tiny evidence, not a proof. This example establishes possibility under explicit assumptions, not that every white-paper observation has that evidential role.

The appropriate answer is therefore conditional: mere logical instantiation does not settle the support supplied by observing this paper. State the background and sampling procedure. A familiar paper deliberately selected as a known nonraven normally supplies no new evidence about raven colour; a specified alternative sampling model may yield a small positive update.

The covering-law model explains a phenomenon by subsuming it under scientific laws and specified circumstances. Its deductive-nomological model requires the event to follow by valid deduction, with a law used essentially, empirical content and true premises. For example, a dynamical law and appropriate initial conditions can explain a body's later trajectory.^[ref]

This captures an important feature of many physical explanations: the event is shown to fit a wider, independently supported pattern. It makes assumptions explicit and connects explanation with prediction. However, derivability alone does not ensure the right explanatory direction. With a pole, a known solar elevation and a measured shadow, geometry lets us calculate either height from shadow or shadow from height. The explanatory asymmetry remains: under the ordinary causal setup, the pole and illumination explain the shadow, rather than its measured length causing the pole's height.

Irrelevant information also exposes a weakness. From a general law plus an unnecessary fact one may still derive the result; logical validity does not show that the added fact helps explain it. Conversely, many accepted explanations are statistical or mechanistic rather than deductions from exceptionless universal laws. Extending the account to the inductive-statistical model helps, but a high conditional probability does not by itself establish causal relevance.

The covering-law model is thus a useful account of one important explanatory structure, not a complete criterion of scientific explanation. A stronger account must consider which assumptions make a difference, what mechanism operates and why the explanatory direction is appropriate. The model's virtues survive when it is treated as a component of explanation rather than an exhaustive definition.

The demarcation problem concerns what distinguishes science from pseudoscience, not merely what distinguishes true claims from false ones. Karl Popper's proposal is falsifiability: a scientific claim must exclude some possible observations, so that a test could in principle count against it. A theory that is compatible with every outcome gains no empirical credit simply by accommodating what occurred.^[ref]

This captures an important methodological defect of pseudoscience: treating every apparent counterexample as further support. But bare falsifiability is not sufficient for good science. A sharply testable lucky-number prediction can be unsupported and methodologically irresponsible. Nor does a mistaken testable theory cease to have been scientific when it is rejected. Scientific status and evidential success are different assessments.

Necessity is also more complicated than a literal one-observation refutation rule. Statistical theories permit rare outcomes, and an instrument-dependent prediction tests auxiliary hypotheses as well as the main theory. Some current research proposals have no feasible immediate test, yet belong to programmes trying to derive discriminating consequences. Their status should depend partly on whether that programme produces such tests, rather than on a permanent exemption from empirical accountability.

Thus falsifiability supplies an important constraint on empirical content but not an adequate stand-alone classification of every scientific activity. Reliable experiments, openness to criticism, comparative explanatory success and correction of failed predictions also matter. This does not make demarcation pointless: it explains why an assessment of a practice over time can be more informative than attaching a binary label to one sentence.

Karl Popper's falsificationism treats theories as conjectures to be exposed to serious attempted refutation. A universal claim entails observable restrictions and can conflict with a counterexample, whereas a finite collection of favourable cases does not deductively establish the universal. This emphasizes risky tests and distinguishes surviving criticism from proof. It also avoids pretending that the problem of induction is solved merely by collecting more confirming observations.^[ref]

The main difficulty is moving from this logical asymmetry to a complete account of practice. A prediction usually has the form $(T\wedge A)\Rightarrow E$, where $A$ includes auxiliary hypotheses about apparatus, initial conditions and background physics. Observing $\neg E$ implies $\neg T\vee \neg A$, not specifically $\neg T$. This Duhem problem makes independent measurement calibration and discriminating follow-up experiments essential. An anomalous detector count can reveal a detector fault rather than a new fundamental interaction.

There are also statistical complications. Under a probability model, an unusual observation need not have zero probability; rejection at a chosen significance level is fallible. Repeated severe tests can rationally alter comparative confidence even though they do not prove a theory. A methodology that merely labels all surviving theories unrefuted misses this evidential differentiation.

Falsificationism is therefore valuable as a critical discipline, especially its opposition to adjustments that insulate a theory from every conceivable result. It is incomplete as a rule requiring immediate abandonment after any anomaly, or as a whole theory of warranted acceptance. The better reconstruction combines explicit possible failures, justified auxiliary hypotheses and comparison of competing explanations, while retaining fallibilism about the outcome.

There are good examples of incommensurability when the claim is that some scientific terms and standards cannot be carried unchanged from one framework into another. They do not establish that rival theories cannot be understood, compared or rationally assessed. The strongest cases concern local interrelations among concepts, rather than a total absence of common experience or language.

The chemical transition associated with phlogiston provides one example. In a phlogistic account, a metal could be treated as a calx combined with phlogiston, with calcination involving the removal of that principle. In an oxygen theory of combustion, the relation is reorganized: the metal combines with oxygen to form the calx. Replacing one word with another while leaving every explanatory relation fixed does not translate the earlier system into the later one. The classifications of elements, compounds and principles are implicated together.^[ref]

This illustrates semantic dependence within a conceptual network. It does not entail that historians cannot reconstruct earlier claims. They can learn how the older vocabulary was used, describe the associated operations and compare its expectations with experimental records. Thomas Kuhn's later treatment distinguishes such interpretation and language-learning from supplying a sentence-by-sentence translation into an unchanged later vocabulary. Explaining the earlier practice need not produce a modern expression which preserves every inferential role of each old term. The fact that the example can be discussed is therefore not, on its own, a refutation of this more restricted thesis.^[ref]

A second example is the changing classification of planets. An Earth-centred arrangement can classify the Sun and Moon with the wandering celestial bodies while excluding the Earth. A heliocentric model reorganizes those relationships: the Earth is a planet and the Moon orbits it. A word-by-word substitution loses a change in the categories and relations used to frame the problem. Yet the investigators can still share observations of apparent positions and disagree about how to explain them. A change in classification is not proof that they inhabit incomparable worlds.

The comparison of Newtonian and relativistic momentum is particularly useful for limiting the claim. With invariant rest mass $m_0$, modern special relativity gives $p=\gamma m_{0}v$, whereas Newtonian mechanics uses $p=mv$. Here $c$ is the speed of light and $\gamma$ is the Lorentz factor. For $v/c\ll 1$ and matched rest-mass convention,This supplies a precise comparison of predictions within a domain. It does not derive all the older interpretation from the newer one with every concept unchanged. Conversely, it would be false to claim that relativity universally replaces invariant mass by a velocity-dependent quantity: invariant rest mass is a standard part of the relativistic account. The historical use of relativistic mass must be distinguished from this convention. For example, at $v=0.6c$, the invariant rest mass is still $m_0$ while $p=0.75m_{0}c$; defining relativistic mass as $p/v$ would give $1.25m_0$. These conventions must not be silently identified. A changed conceptual setting and a controlled mathematical relation between models can coexist.

Methodological differences supply another sense of incommensurability. Investigators may disagree about which problems are central, how much explanatory mechanism is required or how competing virtues should be weighted. Such disagreement can obstruct an algorithmic verdict without abolishing shared values such as accuracy or consistency. Shared criteria may themselves be applied differently. Neither fact demonstrates that no reasons can be offered across the disagreement.^[ref]

The conclusion is accordingly local and discriminating. Chemical kinds, astronomical categories and physical quantities can change their roles in ways not captured by a simple dictionary of synonyms. These examples challenge an account of progress as the addition of truths expressed in an eternally fixed vocabulary. They do not establish wholesale untranslatability, the impossibility of learning a rival framework, equal evidential merit or the irrationality of theory choice. To defend a stronger conclusion one would need to identify the precise comparison that fails and explain why available interpretation, common observations and domain-limited mathematical correspondences cannot repair it.

The history of science is relevant wherever a philosophical claim concerns how inquiry actually develops, what its methods accomplish or what successful theories warrant us in believing. It cannot by itself determine every normative question about rationality. A historical event supplies evidence to an argument; it is not a rule that whatever scientists did must have been rational.

Thomas Kuhn's The Structure of Scientific Revolutions by Thomas Kuhn makes this relevance explicit. Its historical approach challenges the picture of development as the straightforward accumulation of facts and theories under an unchanging method. Investigations conducted within a shared framework can concentrate on puzzle-solving rather than on constant attempts to overthrow their foundations. Episodes of larger change can alter the relevant problems and standards. This is evidence that a philosophical account purporting to describe all successful inquiry must accommodate more than an isolated hypothesis confronted by an isolated observation.^[ref]

The force of a historical example depends on the claim being tested. If a methodology says that rational scientists must reject a theory at its first anomalous reading, a well-documented case of productive work through anomalies is a serious objection to its descriptive adequacy and possibly its practical advice. It does not refute the validity of a deductive rule. From a failed prediction one still infers that some element of the predictive package is wrong; history helps show why identifying that element is not automatic. The Duhem problem supplies the logical distinction, while historical investigation identifies how calibration, alternative explanations and further experiments were actually handled.

History is also central to the pessimistic induction. If some once-successful theories contained rejected posits, predictive success does not guarantee that every theoretical term refers. The replacement of phlogiston theory by an oxygen theory of combustion is philosophically relevant precisely because it changed explanatory roles and material accounting, rather than simply replacing an unpopular name. A responsible comparison examines what each version predicted, how it was modified and which experimental findings were decisive. It should not attribute one conveniently absurd doctrine to every earlier investigator.^[ref][ref]

A realist can respond by distinguishing the parts responsible for success from other commitments or by identifying structure retained across change. This response itself has historical content. It must show the connection between the retained relations and actual achievements, rather than drawing a retrospective line through whatever resembles the present theory. History can constrain both the objection and the defence. It does not uniquely choose between realism, structural realism and empiricism without further epistemic premises.

There are limits to the evidential role. A logical objection to an invalid inference does not require counting how many famous investigators used it. Nor does a prevalence of a practice establish its justification. One must distinguish an account of why a community accepted a view from an argument that the view was supported by the evidence. The context of discovery and the context of justification can be distinguished analytically without assuming that the circumstances of discovery never supply information relevant to justification.

Historical work also requires interpretive care. Sources written for patrons, students or critics can present an ideal of method rather than a complete record of practice. Later terminology can conceal changes in a term's role. Selecting only the eventual winners encourages historical presentism and an exaggerated appearance of inevitability. These are reasons to improve the source-based inquiry, not reasons to ignore it.

The qualified answer is therefore affirmative. History provides counterexamples, identifies live alternatives and tests sweeping pictures of progress, method and theoretical commitment. Philosophy helps formulate what would count as a counterexample and which conclusions the evidence supports. Neither discipline replaces the other. Their interaction is most productive when an explicit general claim is confronted with a carefully reconstructed case, leaving the normative or logical steps visible rather than treating a historical narrative as a self-sufficient verdict.

The raven paradox combines a plausible equivalence principle with an overly unrestricted account of positive-instance scientific confirmation. Let $R(x)$ and $B(x)$ mean that $x$ is a raven and black respectively. ThenA white shoe is a positive instance of the right-hand generalization. If every positive instance confirms its universal generalization, and logically equivalent statements receive the same evidential support, inspecting the shoe seems to confirm the original raven claim. There is no contradiction in the equivalence itself. The tension concerns the evidential rule and the intuition that an arbitrary shoe teaches us nothing about birds.^[ref]

The right response is to make background knowledge and sampling explicit. Under Bayesian confirmation, evidence increases support for $H$ precisely when its likelihood is greater under $H$ than under the alternative, assuming nondegenerate prior probabilities. A fact can therefore be compatible with $H$ yet provide no new support. The mere existence of a white shoe does not determine the needed likelihood ratio.^[ref]

Consider a deliberately simple finite model. There are $N$ objects, exactly $r$ ravens, and $M=N-r$ nonravens, with $M>r>1$. All nonravens are white. Under $H_0$ all ravens are black; under $H_1$ exactly one raven is white. Draw uniformly from the entire population. The event of drawing a white nonraven has probability $M/N$ under either hypothesis. Its Bayes factor is one and it leaves the prior odds unchanged. A black raven instead has likelihoods $r/N$ and $(r-1)/N$, so gives Bayes factor $r/(r-1)$ for $H_0$.

Now change the protocol: draw uniformly from the white objects, then discover that the selected object is a nonraven. This result has likelihoods $1$ and $M/(M+1)$. It gives the small positive Bayes factorFor $N=1000$ and $r=10$, this is $991/990$, whereas sampling a black raven from the full population gives $10/9$. The white nonraven can now help because the search could have revealed a white raven. The two experiments have different selection information, so their differing evidential values do not violate logical equivalence.

This model establishes possibilities, not a universal numerical law about shoes. Different alternatives, population information or selection procedures change the result. If the shoe's being a nonraven was already known, discovering that it is white need not add anything. Conversely, ruling out a potential counterexample can supply a small amount of information. Retain logical equivalence, but reject the unconditional instance-confirmation rule. Once the complete evidential situation is specified, neither zero support nor weak positive support is paradoxical, and neither amounts to proof of a universal law.

Nelson Goodman's new riddle of induction asks which descriptions of an observed regularity are fit for projection. David Hume's problem of induction asks how a projection beyond experience can be justified at all. The problems are related, but accepting that experience supports some expectations does not settle which predicates an acceptable inference may project.

Fix a future time $t$ and let $E_t(x)$ mean that $x$ was examined before $t$. Define grue bySuppose every emerald examined so far is green. Each is then also grue. The same positive instances fit both a green generalization and a grue generalization, although the latter assigns blue to an emerald first examined after the cutoff. This does not require an already examined green emerald to change colour at $t$: its classification uses whether it was examined before $t$, not whether the present time is before $t$. The problem concerns incompatible projections to unexamined objects.^[ref]

It does not resolve the difficulty to object that grue is artificially disjunctive. Introduce the converse predicate, bleen, for examined-before-$t$ blue objects and other green objects. Green can then be defined using grue, bleen and $E_t$. Syntactic simplicity depends on which predicates are primitive. Nor does calling green a genuine natural kind settle matters unless one explains how the relevant natural structure is identified without simply presupposing the preferred projection. Evidence about stable material composition can provide a scientific reason to project a colour response, but that reason goes beyond counting instances.

David Hume's argument exposes a different gap. The denial of an unobserved continuation is not a contradiction. An appeal to the past success of extrapolation already extrapolates from past cases, so cannot supply a noncircular demonstration of its general reliability. His discussion of necessary connection similarly distinguishes experienced successions from an independently perceived necessity binding events together. These claims concern the basis of causal and predictive inference, not merely the spelling of a predicate.^[ref]

Predicate entrenchment is Nelson Goodman's proposed way of distinguishing established projective practices. Green has a history of actual use in projections which its newly constructed rival lacks. This gives a practice-based account of projectibility, rather than an unconditional guarantee that familiar classifications are true. Its weakness is that an account must still permit scientifically successful innovations and correction of entrenched mistakes.^[ref]

Thus Nelson Goodman sharpens the problem facing scientific confirmation: a purely formal positive-instance rule is too permissive. A defensible response combines explicit background assumptions with revisable standards for selecting predicates. That explains why two syntactically parallel generalizations need not receive equal support; it does not provide the noncircular guarantee demanded by the strongest reading of David Hume's challenge.