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Evidence $E$ confirms $H$ relative to background $K$ when it increases its conditional probability. With $0<P(H\mid K)<1$ and positive evidence probability, Bayes' theorem givesIndeed, writing $p=P(H\mid K)$, $a=P(E\mid H,K)$ and $b=P(E\mid \neg H,K)$ gives the posterior probability $pa/(pa+(1-p)b)$; comparison with $p$ reduces to $a>b$. Thus scientific confirmation depends on the alternatives and background, not just on a statement's logical form.

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.

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.

Bayes' theorem solves a precise part of the scientific confirmation problem: it gives a coherent relation between prior commitments, evidential likelihoods and updated probabilities. It does not select those commitments from no premises, and it should not be identified with every meaning of confirmation.

For background information $K$, a hypothesis $H$ and newly considered evidence $E$, let $p=P(H\mid K)$, $a=P(E\mid H,K)$ and $b=P(E\mid \neg H,K)$. With a positive denominator,The first formula requires a positive denominator; the odds formula additionally assumes $0<p<1$ and $b>0$. The second equation separates prior odds from the Bayes factor. For $0<p<1$, the evidence raises the probability of $H$ exactly when $a>b$. It is not enough that $E$ is likely under $H$; it must be more likely than under the relevant alternative mixture. This guards against an inverse-probability error in evaluating scientific tests.

For example, let the prior be 0.01 and the two likelihoods be 0.8 and 0.08. The Bayes factor is ten, but the posterior probability is only $10/109\simeq 0.0917$. The evidence strongly favours $H$ relative to its previous odds without making $H$ probable in an absolute sense. Evidence can therefore confirm a hypothesis while leaving a rational investigator doubtful. The comparison also depends on which alternatives are included in $\neg H$ and how their probabilities are distributed.

The raven paradox shows another benefit. Logically equivalent claims receive identical probabilities in the same probabilistic model, but this does not force every investigation of a white object to provide the same evidence about ravens. If an object is selected because it is already known to be a shoe, observing that it is not a raven may have equal likelihood under the rival hypotheses and provide a Bayes factor of one. Random selection from a specified population can give different likelihoods. The sampling procedure and background information must be represented, rather than letting a grammatical positive-instance rule determine evidential support.^[ref][ref]

The problem of old evidence is a genuine limit of the simplest updating account. If $E$ already belongs to $K$, conditioning on it again produces no increase. Yet an old measurement can be important support for a newly formulated theory. One may assess support relative to a specified earlier or counterfactual background, or model learning a previously unrecognized connection between theory and evidence. These are substantive additions. Simply applying the same conditioning rule to an already certain fact does not represent the new intellectual achievement.^[ref]

Other limitations concern model construction and priors. A hypothesis assigned zero prior probability remains at zero after ordinary conditioning on evidence of positive total probability. A badly specified likelihood can generate confident but unreliable results. Different reasonable priors need not disappear under finite or non-discriminating evidence. Nor does the theorem supply a noncircular solution to the problem of induction: the model's connection to future physical events remains an empirical commitment.

Thus the Bayesian framework gives a rigorous account of probability change, comparison of alternatives and some apparent paradoxes. These are real achievements. It is incomplete as a general recipe for scientific judgment unless the hypotheses, backgrounds, likelihoods and forms of learning are also defended. A mathematical rule of coherent updating can organize evidential reasoning without establishing that every coherent input represents nature correctly.

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.

A commitment to testing claims against experience is a defensible scientific norm. A strict account according to which every meaningful scientific statement must be conclusively verified, or translated into a fixed vocabulary of theory-free observations, is much harder to sustain. Logical empiricism should also be understood as a changing family of projects, not reduced without qualification to its strongest early formulations.

An unrestricted universal scientific law concerns cases beyond a finite observation record. No finite list of favourable cases deductively entails all of them. Requiring conclusive verification would therefore exclude much ordinary theoretical science. Replacing that demand with scientific confirmation retains a relation to evidence while accepting that support is fallible. This is a substantive revision, rather than proof that universal laws were conclusively verified after all.^[ref]

A second difficulty concerns observations. A detector count relies on assumptions about sensitivity, background and the production of a signal. Theory-ladenness makes a sharp division between completely neutral observations and theoretical claims implausible. But dependence on some assumptions does not imply dependence on the truth of the target theory. A calibration can be supported independently, and agreement across instruments using different mechanisms can make a putative effect less vulnerable to one shared error.

Nor should unobservable vocabulary be dismissed merely because its referents cannot be seen directly. A theory of electrons and electromagnetic fields can organize measured currents, spectra and scattering through a common model. What matters scientifically is how the theoretical commitments constrain observations and survive attempts to discriminate them from alternatives. Insisting that each term independently translate into an observation report can miss the role of a whole theoretical structure.

An empiricist can accept much of this. Constructive empiricism permits literal theoretical discourse and commitment to using a theory while restricting belief to empirical adequacy. It separates withholding a stronger ontological belief from refusing to reason with theoretical models.^[ref] Whether this restraint is preferable to selective scientific realism remains an argument about what successful testing warrants.

The strongest sustainable empiricism is consequently methodological and fallible: demand explicit empirical consequences, scrutinize calibration and preserve the possibility of correction. The stricter verificationist reduction does not adequately describe how laws, theoretical terms and measurement actually work. Rejecting that reduction need not weaken the central demand that scientific claims remain accountable to evidence.

Scientific confirmation is evidential support rather than deductive proof. Two distinct problems concern, first, what counts as relevant positive evidence and, second, which part of a tested theoretical package that evidence supports. A defensible response makes background information and the production of evidence explicit. The following two problems are independent: resolving the logical puzzle about positive instances does not eliminate ambiguity about auxiliary hypotheses.

Problem one: the raven paradox. Write the hypothesis aswhere $R$ means “is a raven” and $B$ “is black”. Its contrapositive isThe hypotheses are logically equivalent. A plausible equivalence principle says that identical information must support them equally. A plausible instance principle says that an observed $F$ that is $G$ confirms “every $F$ is $G$”. Combining the principles makes a nonblack nonraven, such as a white shoe, support the raven hypothesis. That conflicts with the intuition that inspecting shoes tells us nothing about ravens. Hempel develops the puzzle in his analysis of instance-based scientific confirmation.^[ref]

My preferred response retains logical equivalence while rejecting an unconditional instance rule. By Bayes' theorem, for evidence $E$, background $K$ and positive relevant probabilities,An instance's bearing depends on how likely it was under the competing possibilities, including how the object was selected. No rule depending only on the grammatical form of an instance can settle that comparison. Branden Fitelson and James Hawthorne analyse the role of background assumptions and Bayesian confirmation in this puzzle.^[ref]

Here is an explicit illustrative model. There are 100 labelled objects: ten known ravens and ninety known nonravens. Under $H$ all ten ravens are black; under the sole alternative $A$, exactly one of the ten is nonblack, with its identity uniformly distributed. Nonraven colours are already known and identical under both possibilities, and the prior probabilities of $H,A$ are equal. Selecting a known white shoe according to a rule independent of $H,A$ has the same likelihood under each, so the posterior probability of $H$ stays $1/2$. A uniformly selected raven, however, is black with probability $1$ under $H$ and $9/10$ under $A$. After observing it to be black,If sampling is instead from known nonblack objects whose membership is uncertain, discovering a nonraven can rule out a possible counterexample and provide support. There is no context-independent prohibition on support from a nonraven. The response explains both the familiar irrelevance intuition and why it is not universally correct. Its limit is that the prior probabilities and sampling assumptions need independent defence; the calculation is a transparent model, not a theorem that all shoes are irrelevant.

Problem two: the Duhem problem of testing a package. Suppose the target hypothesis $T$ and auxiliary hypotheses $A$ jointly imply an outcome $E$:A failure gives $\neg (T\wedge A)$, equivalently $\neg T\vee \neg A$, rather than $\neg T$ alone. Success also does not establish either premise: that would affirm the consequent. Pierre Duhem's point is especially pressing for physical experiments whose predictions require theories of the apparatus as well as the target system.^[ref]

My preferred response combines independent calibration, deliberately varied tests and explicit comparison of packages. For a simple model, suppose a detector has independently calibrated false-positive probability $f$ and detection efficiency $d$. A target predicts a signal on each trial with probability $s$. Then the positive-output probability isUnder a no-signal rival it is $p_0=f$. If $s=1/2$, $d=9/10$, and $f=1/10$, then $p_T=1/2$. One positive reading supplies a Bayes factor of $5$ for the specified target package against the specified no-signal package. But an uncalibrated faulty detector with false-positive probability $1/2$ fits the same positive-reading rate. Target data alone do not distinguish those packages.

Signal-absent calibration trials test the false-positive assumption directly. If ten independent such trials are all negative, their likelihood ratio in favour of $f=1/10$ against $f=1/2$ isThis does not magically isolate $T$: the calibration relies on its own assumptions, and correlated trials would invalidate the power calculation. It shows concretely how additional evidence can discriminate explanations that the initial data leave unresolved. Repeating the target measurement with apparatus based on different physical processes further reduces dependence on one shared failure mode.

Thus the Duhem problem defeats automatic attribution of support or failure, not rational experimental comparison. Where rivals remain empirically equivalent, suspension of judgement about their difference is appropriate. Together, the two responses defend scientific confirmation as a structured, fallible comparison using background information, rather than either pure instance counting or a mechanical verdict on an isolated hypothesis.

Distinguish a criterion of empirical meaningfulness from a methodological demand for verification, and conclusive verification from fallible scientific confirmation.