When Einstein's general theory of relativity was published in 1915, its capacity to explain the anomalous perihelion precession of Mercury—a phenomenon known and unexplained since Le Verrier's calculations in 1859—was widely regarded as powerful evidence for the theory. Physicists treated this as confirmation. Philosophers of science, however, encountered a puzzle.

On the standard Bayesian account of confirmation, evidence E confirms hypothesis H when the posterior probability P(H|E) exceeds the prior P(H). But if E was already known—if the agent already assigned P(E) = 1—then by the axioms of probability, P(H|E) must equal P(H). The evidence confirms nothing.

This is Clark Glymour's problem of old evidence, and it strikes at the heart of probabilistic confirmation theory. Either our most paradigmatic cases of scientific confirmation are illusory, or our formal account of confirmation is incomplete. The puzzle exposes a deeper tension between idealized Bayesian agents, who are logically omniscient and update only on novel propositions, and actual scientific practice, where theoretical novelty and evidential novelty come apart routinely. What follows examines the formal structure of this difficulty, the counterfactual strategies proposed to resolve it, and what historical episodes reveal about how scientists themselves navigate the gap.

The orthodox Bayesian conception of confirmation rests on conditionalization: an agent's credence in H upon learning E should equal the prior conditional probability P(H|E). Confirmation occurs when P(H|E) > P(H), which by Bayes' theorem requires P(E|H) > P(E).

The problem becomes immediate when we consider evidence whose probability is already 1. If E is fully entrenched in the agent's belief system—Mercury's perihelion shift, the photoelectric effect, the cosmic microwave background—then P(E) = 1, P(E|H) ≤ 1, and the likelihood ratio collapses. The mathematics yields zero incremental confirmation, regardless of how elegantly the new theory accommodates the phenomenon.

Worse, the problem extends beyond strict certainty. Any evidence assigned high probability before H's introduction will provide proportionally diminished confirmation, even when H entails E deductively. The very evidence scientists find most compelling—well-established empirical regularities that a new theory unifies or explains—becomes evidentially impotent on a strict Bayesian reading.

This is not a minor technical wrinkle. It implicates the entire predictive/accommodative asymmetry debate and threatens the descriptive adequacy of Bayesianism as a theory of scientific reasoning. If Einstein gained nothing from Mercury, then Bayesian confirmation theory fails to model what physicists in 1915 were manifestly doing.

The diagnosis points toward an idealization in Bayesian epistemology: the assumption of logical omniscience. Real agents do not antecedently know all logical and mathematical consequences of their hypotheses. The discovery that H entails E is itself a cognitive event—and arguably the locus where confirmation actually occurs.

Takeaway

Confirmation may not be a relation between evidence and theory simpliciter, but between evidence and the recognition of an entailment. What changes when a theory is proposed is not the world but our awareness of logical structure.

Daniel Garber and Richard Jeffrey independently proposed a strategy that has become canonical: relocate the probabilistic update from learning E to learning that H entails E. The relevant question is not P(H|E) but P(H|H⊢E)—the credence in H upon discovering the logical relation.

On a related counterfactual rendering, we ask: what would the agent's credence in H have been had they not known E? If P*(H|E) > P*(H), where P* is the counterfactually reconstructed prior with E bracketed, then confirmation obtains. The intuition tracks scientific practice: physicists in 1915 effectively asked how plausible relativity would seem to a hypothetical predecessor who lacked Mercury's anomaly.

Both strategies face technical difficulties. Counterfactual probabilities are notoriously underdetermined—removing E from one's belief corpus requires deciding what else to remove, since beliefs form holistic networks. The probability of Mercury's anomaly is entangled with Newtonian mechanics, observational astronomy, and broader cosmological commitments. There is no unique P minus E.

The Garber-Jeffrey approach fares better technically by treating logical learning as genuine learning within an expanded probability space. But it requires abandoning standard probability axioms, since the agent must assign non-trivial credences to logical truths like "H entails E" before recognizing them. This concession to non-ideal agents is philosophically substantive: it acknowledges that Bayesianism, to be descriptively adequate, must accommodate cognitive finitude.

What emerges is a picture of confirmation as fundamentally diachronic and computational rather than purely structural. Confirmation tracks not the static relation between propositions but the dynamic process by which agents come to apprehend such relations.

Takeaway

When formal theories conflict with manifest practice, the resolution often lies in identifying which idealizations the formalism imposes. Bayesianism's logical omniscience assumption is doing more work—and causing more trouble—than it first appears.

Examining how scientists actually deploy old evidence reveals a sophistication that formal accounts struggle to capture. The acceptance of general relativity drew on Mercury's perihelion, but also on the predicted (and later observed) bending of starlight during the 1919 eclipse. Both functioned evidentially, but in subtly different roles.

Mercury served as a recovery condition: any acceptable successor to Newtonian gravity had to reproduce this anomaly within the theory's natural parameters, without ad hoc adjustment. The eclipse observation served as genuine prediction. Scientists weighted these differently but accorded both substantial evidential force, suggesting the predictive/accommodative distinction is less categorical than philosophers often assume.

Consider also the role of old evidence in unification arguments. Maxwell's electromagnetic theory unified electricity, magnetism, and optics—phenomena all well-known prior to his synthesis. The confirmation derived not from any individual phenomenon but from the structural fact that a single framework subsumed them. Bayesian likelihood ratios applied phenomenon-by-phenomenon fail to register this holistic evidential virtue.

Scientific practice also distinguishes between use-novelty and temporal novelty. A theorist who knows E but does not use E in constructing H may legitimately treat E as confirming evidence, even though E is temporally prior. This nuance, developed by John Worrall and others, suggests that the methodologically relevant question concerns the heuristic path to the theory, not chronology.

These observations vindicate a naturalistic methodology: philosophical accounts of confirmation must be calibrated against detailed reconstruction of how science actually proceeds. Formal frameworks earn their keep by illuminating practice, not by legislating against it.

Takeaway

Scientific reasoning is richer than any single confirmation schema captures. Evidence functions in multiple modalities—recovery, prediction, unification—and a mature epistemology must accommodate this plurality rather than collapse it.

The problem of old evidence is not merely a technical embarrassment for Bayesianism but a productive philosophical lens. It exposes the gap between idealized rational agents and actual epistemic practice, and it forces us to take seriously the cognitive reality of logical discovery.

Whether one favors Garber-Jeffrey solutions, counterfactual reconstructions, or pluralistic accounts of evidential roles, the underlying lesson is that confirmation theory must be psychologically and historically informed. The naturalistic philosopher treats formal epistemology not as a priori legislation but as an ongoing dialogue with science as practiced.

What looks like a paradox about probability turns out to be a window onto something deeper: the structure of theoretical insight itself. Recognizing that a theory entails what we already knew is, in its own right, a genuine cognitive achievement—and perhaps the truest measure of explanatory power.