A NEW GARBER-STYLE SOLUTION TO THE PROBLEM OF OLD EVIDENCE - Stephan Hartmann and Branden Fitelson

Oct 22, 2026

Uploaded by ahduag

A Bayesian confirmation-theoretic framework is developed to address the “old evidence” problem concerning Einstein’s credences about Mercury’s perihelion. The text reviews Garber-style models, where learning X supports the target hypothesis H by linking it to evidence E through entailment-style constraints, then critiques Jeffrey’s and Earman’s reliance on strong assumptions about E and on interpreting X and Y as “H entails E” versus “H refutes E.” A new approach replaces entailment with explanatory adequacy, using X as adequate explanation of E and Y as the best competitor’s adequate explanation, aiming to relax prior certainty and entailment interpretations.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages3

Reading Time8 min

AccessFree to download

Background: Garber, Jeffrey and Earman

Garber-style Bayesian models and constraints

Jeffrey’s disjunction constraint

Earman’s ordinal constraint and alternative approach

A New Garber-Style Approach

Explaining old evidence via explanatory adequacy

A NEW GARBER-STYLE SOLUTION TO THE PROBLEM OF OLD EVIDENCE STEPHAN HARTMANN AND BRANDEN FITELSON 1. Background: Garber, Jeffrey and Earman By the time Einstein had formulated the general theory of relativity (H), the evidence regarding the perihelion of Mercury (E)— which Newtonian theory was unable to adequately explain — had long been known (Roseveare 1982) . Indeed, it is not implausible to suppose that Einstein was certain (in 1915) that E was true. Nonetheless, it is widely accepted that Einstein learned some proposition X (in 1915) which had the eﬀect of conﬁrming H (i. e. , rationally raising Einstein's credence in H) . Garber (1983) proposes that what Einstein learned was a logical fact (i. e. , that X 􀂃 “H entails E”) . By adding an additional atomic statement “X” to the H; Elanguage (and interpreting “X” extra-systematically as “H entails E”), Garber showed how it was possible to write down Bayesian models of this sort, having the following desired conﬁrmation-theoretic property (y) Pr􀂄H j X􀂅 > Pr􀂄H􀂅: Garber did not, however, endorse speciﬁc constraints on Pr􀂄􀀁􀂅, which ensure (y) . Subsequently, Jeﬀrey (1992) and Earman (1992) oﬀered such constraints. Their approaches require the addition of X and a fourth atomic statement “Y” to the H; E-language, where Y is extra-systematically interpreted as “H entails :E” (viz. , H refutes E) . In these traditional Garber-style approaches, the background extrasystematic constraints on Pr􀂄􀀁􀂅 consist of the following pair of “modus ponens principles” for the extra-systematic entailment relation.1 (1) Pr􀂄E j H & X􀂅 􀂃 1 (2) Pr􀂄E & H & Y 􀂅 􀂃 0 In addition to (1) and (2), Jeﬀrey oﬀers the following extra-systematic constraint: (3) Pr􀂄X _ Y 􀂅 􀂃 1 Informally,(3) expresses certainty in the disjunction: either H entails E or H refutes E. Given the background assumption that Pr􀂄E􀂅 􀂃 1, it is straightforward Date: 06/10/15 . Draft: Final version to appear in Philosophy of Science. We would like to thank Fabrizio Cariani, Vincenzo Crupi, Kenny Easwaran, Aidan Lyon, Jan Sprenger, Mike Titelbaum and two anonymous referees for useful feedback on previous versions of this paper. 1Traditionally, Garber-style approaches also rest on the background assumption that Pr􀂄E􀂅 􀂃 1. In standard probability theory, this background assumption entails (1) . We have chosen to state (1) explicitly here, since it reﬂects the fact that X is (in traditional Garber-style approaches) couched in terms of entailment. In the next section, we explain how to relax both of these assumptions. 2 STEPHAN HARTMANN AND BRANDEN FITELSON to show that (1)–(3) jointly entail the desired conﬁrmation-theoretic conclusion (y) . This, in essence, was Jeﬀrey's Garber-style approach to the old evidence problem. As Earman rightly points out, Jeﬀrey's constraint (3) is not very plausible as a rational stricture on Einstein's credences, prior to his learning X (in 1915) . There was no good reason for Einstein to be certain (prior to learning X) that H either entails or refutes E. Earman oﬀers the following alternative extra-systematic constraint: (4) Pr􀂄H j X􀂅 > Pr􀂄H j :X & :Y 􀂅 Earman shows that — given the background assumption that Pr􀂄E􀂅 􀂃 1 —(1), (2) and (4) jointly entail (y); and, he argues that (4) is a plausible assumption regarding Einstein's credences (in 1915) . We agree that (4) is more plausible than Jeﬀrey's (3) . For one thing,(4) is not a numerical constraint, but merely an ordinal constraint on Einstein's 1915 credences. Moreover, because Einstein was (antecedently) certain of E , it is reasonable to suppose that he would have judged that X confers a greater probability on H than :X & :Y does. Having said that, we would prefer a more general approach, which (a) doesn't presuppose that Pr􀂄E􀂅 􀂃 1; and,(b) doesn't require interpreting X and Y in terms of entailment relations. In the next section, we describe just such an approach. 2. A New Garber-Style Approach We like Garber's idea

FAQs

What is the “old evidence” problem addressed in the paper?

The paper focuses on how learning in 1915 could rationally increase Einstein’s credence in the general theory of relativity given prior knowledge of Mercury’s perihelion anomaly.

How do traditional Garber-style approaches interpret X and Y?

They interpret X and Y using entailment relations, treating X as a supporter of H toward E (H entails E) and Y as a refuter (H refutes E), together with extra-systematic probability constraints.

What is the main change in the new Garber-style approach?

X and Y are reinterpreted as explanatory adequacy: X means H adequately explains E, while Y means H’s best competitor adequately explains E. This aims to avoid presupposing Pr(E)=1 and avoid entailment-based interpretations.

Oct 22, 2026

Uploaded by ahduag