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REVIEW 4 major objections 5 minor 59 references

A not-too-simple solution to Goodman's new riddle of induction in the age of AI

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that Goodman's new riddle of induction dissolves once direct measurements are required to yield convex error-boxes and model complexity is measured by the shortest empirically equivalent formulation.

desk verdict A transparent synthesis of Gärdenfors and the author's own 2013 model that makes a real proposal, but the green/grue asymmetry is largely stipulated by Postulate 1, so 'solution' overstates what is demonstrated. read the letter →

arxiv 2507.17212 v1 pith:JCNZOEZM submitted 2025-07-23 physics.hist-ph

Luigi Scorzato This is my paper · ORCID
classification physics.hist-ph
keywords Goodman'snewriddleofinductiongruemodelselectionepistemiccomplexityconvexitydirectmeasurementunderdeterminationKolmogorov-Chaitin
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that Goodman's new riddle of induction—the puzzle of why 'green' is projectible but 'grue' is not—can be solved by tying two ideas together: a direct measurement must report a central value inside a single convex error-box, and a model's complexity is the shortest formulation of its assumptions among all logically and empirically equivalent versions. Under that standard, the familiar $\Xi=0$ trick fails: rewriting every hypothesis as one equation is not empirically equivalent to the original model, because it would require the lumped quantity $\Xi$ to be directly measurable with a connected error-box, which grue-like quantities are not. The paper argues that this yields a simple selection rule—drop any model that is more complex and no more accurate than a competitor—that matches the model choices scientists actually make, and that this descriptive match is what a solution to the new riddle should be. The reason to care, beyond resolving a textbook paradox, is that the same trick now threatens the reliability assessment of AI models.

What carries the argument

The machinery is the pair consisting of Postulate 1 and Definition 3. Postulate 1 requires the outcome of a single direct measurement of a property $Q$ to be a central value $Q_0$ together with a convex error-box containing $Q_0$; it rules out split error regions as legitimate direct outcomes. Definition 3 defines the epistemic complexity $C(M)$ of a model $M$ as the minimum, over all logically and empirically equivalent formulations $M'\equiv M$, of the length of the assumptions $P(M')$; here empirical equivalence (Definition 2) requires the translation to preserve the measurement outcomes and precisions of every measurable property. Together these make the $\Xi=0$ reformulation inadmissible, because that rewriting breaks empirical equivalence and demands a directly measurable $\Xi$ with a connected error-box, so the minimum is non-trivial and reformulation-independent. The paper's one-sentence summary of the mechanism: 'it is the constraint of convexity that enables a non-trivial notion of complexity.'

What would settle it

A single direct measurement whose reported uncertainty is a genuinely disconnected or non-convex region—a multimodal expected distribution for one measurement—would falsify Postulate 1 and collapse the complexity gap, because grue would then be as directly measurable as green. Alternatively, any case of model selection that the broad scientific community rejects but Definition 4 admits, or accepts but Definition 4 rules out, would falsify the paper's descriptive claim.

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Extended reading notes

Core claim

The central claim is that the asymmetry between green and grue is not a matter of language or convention but of measurability: although any model can be expressed as $\Xi=0$, no empirically equivalent formulation can make $\Xi$ a directly measurable quantity with a central value and a connected error-box, because the error-box of a grue measurement splits at the critical time $t_0$ and, for emeralds first seen before $t_0$, remains undetermined later. The paper formalizes this through Postulate 1 (convexity of single direct measurement outcomes) and Definition 3 (epistemic complexity as minimum length over logically and empirically equivalent formulations), and shows that the grue model is then strictly more complex than the green model with no empirical advantage, so Definition 4 rules it out. The paper stresses that this solves the new riddle—identifying the hidden assumptions behind scientists' actual model selection—and not the old riddle of justifying induction by future success.

Load-bearing premise

The argument hangs on Postulate 1, that a single direct measurement always reports a central value inside one convex error-box and never a split or disconnected region; if scientists could legitimately report a split region as a direct measurement, grue would be directly measurable and the complexity advantage of green disappears.

Editorial extensions

If this is right

  • The selection rule of Definition 4 gives a precise meaning to 'explaining more with less': any model that is more complex and no more accurate than a rival is ruled out, with no trade-off involved.
  • Because epistemic complexity is invariant under logically and empirically equivalent reformulations, simplicity comparisons remain meaningful across very different theories, including across scientific revolutions.
  • The same framework dismisses conspiracy theories: extra ad-hoc assumptions like '$\Xi$-people' buy conciseness only by sacrificing empirical accuracy, because the corresponding measurements are not available.
  • Bayesian confirmation requires prior probabilities; the paper argues the prior choice can be anchored only by this reformulation-independent complexity measure, otherwise the $\Xi$ trick makes priors arbitrary.
  • This is a solution to the new riddle, not the old one: it describes the hidden assumptions behind scientists' actual choices rather than promising to justify the future success of science.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extending the paper's approach: any predicate that forces split error-boxes in a shared direct-measurement basis should be empirically disfavored exactly like grue; this is a testable prediction for language design and machine-learning feature engineering.
  • The paper asserts rather than proves that the minimum in Definition 3 is non-trivial and attained; a formal proof, or a realistic model class where the minimum fails to be attained, would either complete or stress the foundation.
  • A broader research program implied here is grounding non-empirical epistemic values (simplicity, naturalness, projectibility) in measurability constraints rather than in metaphysical natural kinds or entrenchment.
  • For AI, the framework suggests a concrete auditing rule: a learned model that gains apparent simplicity by redefining its inputs so that measurement error-boxes split is a 'grue model' and should be downgraded; this could be operationalized as a test for shortcut learning.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a solution to Goodman's new riddle of induction by combining two ideas: (i) a postulate that the outcome of a single direct measurement is always a central value with a convex error region, and (ii) an "epistemic complexity" defined as the minimum length of the assumptions over all logically and empirically equivalent reformulations. The author argues that a grue-like formulation cannot be empirically equivalent to the standard green formulation because measuring grue requires a non-convex, split error-box, so any grue reformulation is more complex without empirical advantage. The paper also argues that this combination solves the riddle in the sense of identifying the hidden assumptions behind scientific model selection, and it offers historical and conspiracy-theory examples as illustrations.

Significance. If the central claims hold, the paper would provide a principled, reformulation-invariant criterion that rules out grue-like predicates without appealing to psychological entrenchment or subjective simplicity, and it would connect a classic philosophical riddle to practical questions in AI model selection. The paper is clearly written, engages seriously with the literature on conceptual spaces and complexity, and explicitly identifies the adequacy conditions for a solution. The main contribution is conditional, however: the two load-bearing assumptions—convexity as a constitutive feature of direct measurements and the non-triviality of the minimum in Definition 3—are asserted rather than demonstrated. The paper's descriptive examples are suggestive but do not yet provide independent evidence for those assumptions.

major comments (4)
  1. [Section 2.1, Postulate 1] The central asymmetry between green and grue rests on excluding split error regions from the outcome of a single direct measurement. The author explicitly acknowledges that Post. 1, together with Def. 1, offers an "implicit (partial) definition of direct measurements" and that "it is up to the model to decide which are the direct measurements." Consequently, the claim that grue is not directly measurable is not an empirical result but a consequence of the postulate; a proponent of grue could simply stipulate that a grue-meter's single-readout distribution is bimodal and insist that it is direct. To carry the argument, the paper needs an independent characterization of directness, or at least a systematic empirical survey showing that no accepted scientific direct measurement has a non-convex error-box. The two examples given (room temperature and Fig. 2) are not enough to support the load placed on this postulate.
  2. [Section 2.2, Definition 3] The epistemic complexity C(M) is defined as a minimum over "all possible equivalent formulations (in any language)" of length[P(M')]. The paper does not specify the length function, does not prove that the minimum exists, and does not prove that it is non-trivial for the relevant cases. The statement that restricting to logically and empirically equivalent formulations ensures that the Xi = 0 formulation is no longer legitimate, and that the shortest formulation is "in general, not trivial anymore," is an assertion. Without a precise definition of the language class and a proof that no short, convex, directly measurable reformulation can encode the same empirical content, the complexity gap between green and grue could collapse, and the definition might be ill-defined or yield a constant for all models.
  3. [Section 2.2, Definition 2] The empirical equivalence relation used in Definition 3 requires "same precision and same outcome" for each measurable property. Precision is not an external fact; it is part of the model assumptions, since Definition 1 includes Delta(b) for every directly measurable quantity b in B. This creates a circularity: whether two formulations are equivalent depends on the very precision values whose effect on complexity is being assessed. The paper should specify how equivalence is to be judged independently of the model's own stipulations, or explain why this dependence does not undermine the claimed reformulation independence.
  4. [Sections 3.2 and 5] The paper's justification strategy is descriptive accuracy: the model is accepted if no counterexample against scientific consensus exists. But the paper never operationalizes "broad scientific consensus" nor explains how to identify a counterexample independently of the framework. The Bielefeld conspiracy example is essentially a restatement of the same unavailability-of-records argument used for grue, so it does not provide independent support. The "no counterexample" claim is therefore a conjecture rather than a test of the model; the author should either provide a falsification protocol or soften the claim accordingly.
minor comments (5)
  1. [Section 2.1, Postulate 1'] The paper says that all important conclusions are maintained under the more general Postulate 1', but it does not demonstrate this; a short verification would be helpful, especially given that the probability-distribution formulation is what makes the convexity claim checkable.
  2. [Section 2.2, Definition 3] The term length[P(M')] is never defined. If it is intended as a string length in some formal language, that language must be specified; if it is an informal notion of amount of assumptions, the claim of precision is strained.
  3. [Throughout] There are several typographical errors, including "Goodnam" in Section 1.2, "constrints" in Section 2, "explicitely" and "alghough" in Section 2.1, "Gardenfor's" in footnote 9, and "discipleines" in Section 5. These should be corrected.
  4. [Figure 3] The right panel of Fig. 3 would be more informative if the axes were labeled explicitly (e.g., wavelength and time), so that the reader can see exactly how the error-box splits in the grue/bleen representation.
  5. [Section 4.2] The discussion of knowledge-what is interesting but seems only loosely connected to the formal definitions in Section 2; the author should state explicitly whether the irreducibility of knowledge-what is supposed to justify Postulate 1 or is merely a philosophical aside.

Circularity Check

1 steps flagged · score 6.0 of 10

The green/grue asymmetry is stipulated by Postulate 1's definition of direct measurements, making the central 'prediction' true by construction.

  1. self definitional [Section 2.1, Postulate 1 and following paragraphs (around Fig. 3)]
    "Postulate 1 The result of a valid single direct measurement of (k-dimensional) property Q is always expressed as a (k-dimensional) central value Q0 and a (k-dimensional) convex set (error-box) that contains Q0. ... Note that I have not defined direct measurement explicitely. Post. 1—together with Def. 1 below—offers an implicit (partial) definition of direct measurements. ... Split error-boxes are incompatible with a direct measurement, although they are fully acceptable for indirect measurement."

    The paper's central claim is that grue is not directly measurable and therefore more complex than green. But 'direct measurement' is not an independent concept; it is stipulated by Post.1 to be convex. The grue error-box (Fig. 3) is non-convex, so 'grue is not directly measurable' is a direct substitution into the definition. The paper even concedes that Post.1 is an implicit (partial) definition. Consequently the asymmetry the solution relies on is built into the premise: a hypothetical grue-meter whose single-readout distribution is bimodal is rejected for no reason other than the postulate. The later complexity argument (Def.3) then inherits this stipulated asymmetry rather than discovering it.

full rationale

Most of the paper is an explicit axiomatic construction rather than a hidden circularity: Post.1 (convexity of direct measurements) and Def.3 (minimal length over empirically equivalent formulations) are stated as postulates/definitions, and the applications (grue, Bielefeld conspiracy) are worked out from them. However, the central asymmetry is partly stipulated. Post.1 is acknowledged to be an implicit (partial) definition of 'direct measurement,' and the paper then uses the non-convexity of the grue error-box to conclude that grue is not directly measurable. That conclusion is a direct consequence of the definition: any proposed bimodal single-readout 'grue-meter' is inadmissible by fiat. Hence the claimed reduction of Goodman's riddle to complexity is conditional on a premise that already encodes the desired distinction. A second load-bearing assertion is Def.3's non-triviality: the claim that restricting to empirically equivalent formulations makes the minimum length 'in general, not trivial anymore' is asserted without proof; if the minimum collapsed to a constant for all models, the complexity gap would disappear. This is an omitted proof rather than a circular step per se. The paper's appeals to Scorzato (2013) for the framework and for the absence of counterexamples are self-citations, but they support applications rather than the core derivation. Overall, the solution has independent, testable content against scientific-consensus examples, so the circularity is partial, not total.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

No numerical free parameters are fitted, but the framework contains several under-specified choice functions: the length measure, the precision Delta, and the metric d. The main conceptual burden is Postulate 1 and the assumed non-triviality of epistemic complexity. No physical entities are posited, but the new measure C(M) is a conceptual invention with no external falsifiable handle.

free parameters (3)
  • Length function in Definition 3 = unspecified
    Epistemic complexity is defined as a minimum length of assumptions, but no precise measure of length over arbitrary languages is given; the non-triviality claim depends on it.
  • Precision Delta(b) for the direct measurement basis B = unspecified
    Definition 1 requires each directly measurable quantity b to be measurable with precision Delta(b), but Delta is assigned by the model and not determined by the framework.
  • Metric d for discrete measurements = unspecified
    Appendix A introduces a metric on discrete outcome spaces that must reflect confusion probabilities, but no constructive rule is given, leaving a degree of freedom in the framework.
assumptions (5)
  • domain assumption A valid single direct measurement always yields a convex error region (Postulate 1/1').
    This is the central stipulation used to make grue non-directly measurable; the paper justifies it by scientific practice, not by derivation.
  • ad hoc to paper The minimum in Definition 3 over all logically and empirically equivalent formulations is non-trivial.
    The paper asserts that empirical equivalence prevents the Xi=0 collapse, but no proof or measurable construction is given.
  • domain assumption For any two models on the same topic, a common sub-model containing all directly measurable quantities exists.
    Section 2.2 dismisses incommensurability as a theoretical possibility with no evidence, but this is an empirical claim about all of science.
  • domain assumption Broad scientific consensus is a reliable benchmark for model selection and for testing the proposed model.
    The definitions of what "should" be ruled out in Section 3 rely on consensus as ground truth, without independent justification.
  • domain assumption The set of measurable quantities can be captured by a finite basis B of directly measurable concepts.
    Definition 1 assumes a basis exists sufficient to operationally define all other measurables; no existence theorem is provided.
invented entities (1)
  • Epistemic complexity C(M)
    purpose: A formulation-independent measure of the complexity of a model's assumptions, used to rule out models that are more complex with no accuracy gain.
    It is a new conceptual quantity, not independently measurable; the paper asserts it is precisely defined but only approximately estimated, and its applicability to new cases is not demonstrated.

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Pith. "Pith review of A not-too-simple solution to Goodman's new riddle of induction in the age of AI." pith.science (2026). https://pith.science/paper/JCNZOEZM

@misc{pith2026250717212,
  author       = {Pith},
  title        = {Pith review of: A not-too-simple solution to Goodman's new riddle of induction in the age of AI},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JCNZOEZM}},
  note         = {Machine review of arXiv:2507.17212}
}
read the original abstract

I review the works of G\"ardenfors (1990) and Scorzato (2013) and show that their combination provides an elegant solution of Goodman's new riddle of induction. The solution is based on two main ideas: (1) clarifying what is expected from a solution: understanding that philosophy of science is a science itself, with the same limitations and strengths as other scientific disciplines; (2) understanding that the concept of complexity of a model's assumptions and the concept of direct measurements must be characterized together. Although both measurements and complexity have been the subject of a vast literature, within the philosophy of science, essentially no other attempt has been made to combine them. The widespread expectation, among modern philosophers, that Goodman's new riddle cannot be solved is not defensible without a serious exploration of such a natural approach. A clarification of this riddle has always been very important, but it has become even more crucial in the age of AI.

Discussion (0). Continue with ORCID to comment.

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Reviewed August 6, 2026 · model on record in the stance chip above.