The use of knowledge in open-ended systems

Abigail Devereaux; Roger Koppl

REVIEW 5 major objections 5 minor 93 references

The use of knowledge in open-ended systems

T0 review · 5 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper argues that in open-ended evolutionary systems, observers' knowledge is local and frame-relative: common knowledge is generally impossible, and rational agents can agree to disagree.

desk verdict A stimulating essay whose formal theorems are not established: the impossibility results rest on an unproved conjecture and several appendix proofs are invalid. read the letter →

arxiv 2412.00011 v1 pith:MFIZ7NZE submitted 2024-11-13 cs.NE cs.AIcs.LOecon.TH

Abigail Devereaux , Roger Koppl This is my paper · ORCID

classification cs.NEcs.AIcs.LOecon.TH MSC 03B4291A26

keywords open-endedevolutionepistemiclogiccommonknowledgeframerelativityagreetodisagreeadjacentpossiblenon-ergodicundecidabledisjunctions

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

The paper tries to show that the standard epistemic-logic machinery used in economics, which assumes a complete, shared state space and common knowledge, breaks down when the system generates genuinely novel possibilities. It builds a formal framework for knowledge use and acquisition in open-ended evolutionary (OEE) systems and claims five results: believed truths never fully coincide with true truths, common knowledge is generally impossible, agents can agree to disagree, nonlogical search can be as knowledge-generative as logical search, and knowledge is non-ergodic. A sympathetic reader would care because these results, if correct, would replace the closed-world assumptions of rational-choice theory with a picture in which each observer is bound to a local, evolving frame and coordination must be achieved through institutions, aesthetics, and other nonlogical codes.

What carries the argument

The framework's load-bearing objects are the observer-specific theory $T_{i,t}$, the model $M_{i,t}$, the set of nonlogical constants $P_{i,t}$, and the decision procedure $\delta_{i,t}:\Omega_{i,t}\to\Sigma_{i,t}$ that assigns truth values to sentences. Local knowledge $\kappa_{i,t}(\omega)$ is the set of sentences in state $\omega$ prefixed with the knowledge operator $k_{i,t}$; the contextual knowledge possible $K_{i,t}$ collects all states decidable by $\delta_{i,t}$; and the adjacent knowledge possible $A_{i,t}=K_{i,t+1}\setminus K_{i,t}$ captures the new states that enter between time steps. The Disjointness Conjecture, that $K_{i,t}\neq K_{j,t}$ in general for different agents, is what carries the impossibility of common knowledge, while the non-convergence of theory revision, grounded in Gödelian undecidability, carries frame relativity.

What would settle it

Construct an open-ended evolutionary system with two observers who, despite continual novelty, maintain identical knowledge sets $K_{i,t}=K_{j,t}$ at every time step, for example by having all new predicates broadcast symmetrically to both; then common knowledge would be possible and OEE2 and OEE3 would fail. Alternatively, exhibit a specific OEE process whose theory revision sequence $\{T_{i,t}\}_{t}$ converges to a decidable $T_\$\Omega$$, which would falsify OEE1 and frame relativity.

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Extended reading notes

Core claim

The central discovery the paper asserts is frame relativity: a complete and correct theory of an open-ended evolutionary system is possible only for an agent outside the system, while any embedded observer is limited to their own known-world $\Omega_{i,t}$, theory $T_{i,t}$, and decision procedure $\delta_{i,t}$. From this, the paper derives five results. OEE1 says an observer's believed-true propositions and the true propositions of the system can never coincide. OEE2 says common knowledge is in general impossible because knowledge hierarchies of different individuals are not mutually consistent. OEE3 says agents can therefore rationally agree to disagree. OEE4 says random, heuristic, or aesthetic search can be as knowledge-generative as logical search because logic alone cannot resolve the undecidable disjunctions an OEE system continually produces. OEE5 says knowledge in such systems is non-ergodic: the time average of local knowledge does not converge to a state-space average because the adjacent possible is always nonempty.

Load-bearing premise

The proof that common knowledge is impossible and that agents can agree to disagree rests on the unproved Disjointness Conjecture, which asserts that different observers' local knowledge sets are at least partially disjoint across time and agents; the proof sketch appeals to the very non-convergence result the paper is trying to establish.

Editorial extensions

If this is right

No observer embedded in an OEE system can ever reach the theory of everything $T_\Omega$, and the number of statements true in $T_\Omega$ but undecidable in $T_{i,t}$ is infinite.
Common knowledge cannot be presumed in open-ended systems, so coordination must be achieved through explicit institutions, shared codes, and other knowledge-sharing mechanisms.
Agents can rationally agree to disagree, overturning the classical result that common priors and common knowledge force agreement.
Nonlogical modes of reasoning such as aesthetics and heuristics are not mere biases; they can be as knowledge-generative as logical deduction when undecidable disjunctions are unavoidable.
Knowledge in OEE systems is non-ergodic, so an individual's time average of knowledge cannot be replaced by an ensemble average over possible states.

Reading between the lines

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

Extension: if frame relativity is correct, expert failure and scientific consensus are not merely corrigible by more computation or better data; they are structural limits of any embedded observer, which would extend Hayek's knowledge problem to organized science.
Extension: the framework suggests a testable prediction that faster open-ended change should increase cultural, ideological, and aesthetic fragmentation, because shared nonlogical codes become more valuable and cheaper to maintain in niches.
Extension: the non-ergodicity result gives a formal epistemology for the recent critique of expected utility theory based on ergodicity; one could test it by simulating an artificial-life system where agents with asymmetric observations of novelty repeatedly disagree despite common priors.
Extension: the 'many-selves' reading of reasoning through time implies that preference change is not just a parameter shift but a change in the observer's known-world, which could be operationalized in behavioral experiments on how people reinterpret past choices after radical novelty.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague—

The headline: this is a stimulating essay wrapped in a formalism that does not hold together. The authors want to formalize the Hayek/Kauffman picture of radically uncertain, open-ended systems, and they write with real clarity about why that picture matters. But the central theorems OEE1–OEE5 are not established; several proofs use the thing they are out to prove.

What is genuinely useful: the distinction between the contextual knowledge possible and the adjacent knowledge possible, and the frame relativity idea, are clean ways to describe why embedded observers cannot have the god's-eye theory of the system. The paper also does a good job situating itself in the OEE literature and in the Aumann–Hintikka tradition. The institutional conclusion—that shared codes emerge as a substitute for impossible common knowledge—is suggestive.

Now the soft spots, and they are not cosmetic. The Disjointness Conjecture (Conjecture 1) is doing all the work, and its proof sketch is circular. It assumes that if two agents share knowledge sets then they share theories, then says theory convergence is impossible by Theorem 1. But Theorem 1 only says an individual's theory cannot in general be an inessential extension of its own past theory when new predicates arrive. It says nothing about whether two agents' theories converge. Open-endedness is defined per individual; it is compatible with every agent seeing the same new predicates and converging to the same theories. So the 'general disjointness' claim is simply unproved. Corollary 2 then starts with 'Ti,t ≠ Tj,t ≠ TΩ in general,' which is the conjecture restated as a premise, and concludes common knowledge is impossible. Corollary 3 asserts agree-to-disagree as a direct consequence, but Aumann's result needs posteriors and common priors; none are defined. Proposition 8 cites a nonexistent 'Theorem 7.' Lemma 2 says that if there were finitely many undecidable disjunctions then Ti,t+τ = TΩ; that does not follow—finitely many may be resolved in τ steps, but the theory may still not be the theory of everything. The non-ergodicity proof also leans on the same nonempty adjacent possible it is trying to establish.

I want to give credit: the paper does not hide these moves, and the prose is honest about the tradition. But the formal machinery is not yet doing the work. If OEE2 and OEE3 were relabeled as conjectures, the paper would be a reasonable essay; as theorems, they fail.

For peer review: I'd send it out, because the conceptual questions are significant and a good referee can pin down exactly what needs proving. But I would not accept it in anything like this form. The authors need to either prove the Disjointness Conjecture, weaken their claims to conjectures, or drop the Aumann 'agree to disagree' language. The paper deserves a reading group debate, and a serious referee, but the referee should be told to go straight for the conjecture.

Referee Report

5 major / 5 minor

Summary. The paper develops a syntactic epistemic-logic framework for knowledge in open-ended evolutionary (OEE) systems. It defines per-agent "known-worlds" Ω_{i,t}, theories T_{i,t}, predicate sets P_{i,t}, decision procedures δ_{i,t}, contextual knowledge sets K_{i,t}, and local knowledge κ_{i,t}, and it proposes five headline results: OEE1 (believed truths never fully coincide with true truths), OEE2 (common knowledge is generally impossible), OEE3 (agents can agree to disagree), OEE4 (nonlogical search can be as knowledge-generative as logical search), and OEE5 (knowledge is non-ergodic). The formal apparatus is presented in an appendix with definitions, propositions, and corollaries. The central philosophical claim is "frame relativity": an observer embedded in an OEE system cannot possess a complete and correct model of the system, and individual knowledge is local and fragmented across agents and time.

Significance. If OEE1–OEE5 were soundly derived, the paper would provide a substantive formal challenge to closed-world epistemic logic and would motivate a research program on institutions, aesthetics, and heuristics as knowledge-generating devices under genuine novelty. The authors are to be credited for making their formalism explicit enough that the gaps are checkable, and for transparently labeling the key cross-agent assumption as a conjecture. However, the significance is currently prospective: the individual-level incompleteness results largely restate the definition of open-endedness, while the cross-agent results OEE2 and OEE3 rest on an unproved conjecture and an undeveloped probabilistic model. The paper supplies no machine-checked proofs and no falsifiable predictions, and several appendix proofs are invalid as written. The central claims are therefore not established by the manuscript in its present form.

major comments (5)

[§3.2, Conjecture 1; Appendix, Proposition 6 and Corollary 2] The proof of OEE2 uses the unproved Disjointness Conjecture as a premise. Corollary 2 begins "Under open-ended evolution, T_{i,t} ≠ T_{j,t} ≠ T_Ω in general," which is exactly the conjecture restated as a fact rather than a derived conclusion. The proof sketch of Conjecture 1 appeals to Theorem 1, but Theorem 1 concerns a single agent's theory extension across time (P_{i,t+1}\P_{i,t} ≠ ∅) and says nothing about whether two different agents' theories or knowledge sets are disjoint. Definition 7 is an individual condition; it is compatible with all agents observing the same sequence of new predicates and sharing identical theories and known-worlds at every time. Therefore OEE2 is not established, and Proposition 6's claim that hierarchies are "in general" inconsistent inherits the same gap.
[Appendix, Corollary 3] The "agree to disagree" conclusion does not follow from Proposition 6 and Corollary 2 as stated. In Aumann's sense, agreeing to disagree means that differing posterior beliefs are common knowledge, which requires a probabilistic apparatus: states with prior probabilities, posterior belief functions, and a definition of common knowledge of those beliefs. The paper never defines posteriors, common priors, or a probabilistic counterpart of K_{i,t}. Corollary 3 is asserted as a "direct consequence" without supplying any of the machinery that gives the phrase its standard meaning, so OEE3 is not demonstrated.
[Appendix §6.7, Proposition 8 and Corollary 5] Proposition 8's proof cites "Theorems 7 and 5," but no Theorem 7 exists in the manuscript. More substantively, the argument that an algorithmic process m "does not halt" does not establish that there exists a sentence ξ' in Ω for which m is not specified: a non-halting process could still decide every sentence at some finite stage. Corollary 5 then infers from Proposition 8 that T_{i,t} is not recursively enumerable, which is a much stronger claim than the proposition establishes. Thus the frame-relativity result, as formalized, is not proved.
[Appendix, Lemma 2] The finite-τ argument is logically invalid. From the assumption that there are finitely many undecidable disjunctions D_{i,t} with |D_{i,t}| = τ, the proof concludes that T_{i,t+τ} = T_Ω. But open-endedness only guarantees that new predicates arrive at each step; it does not imply that the number of undecidable disjunctions equals the number of steps to the theory of everything, nor that T_Ω is reached after τ steps at all. The further claim that D_{i,t} is uncountably infinite is also unsupported by the preceding argument, which at most suggests an infinite set.
[Definition 7, Proposition 5, Lemma 1, Theorem 2] The individual-level impossibility result OEE1 is true by construction rather than by substantive proof. Definition 7 defines open-endedness as P_{i,t+1}\P_{i,t} ≠ ∅; Proposition 5 derives the nonemptiness of the adjacent knowledge possible A_{i,t} from this definition; Lemma 1 derives the existence of an undecidable sentence; and Theorem 2 uses Lemma 1 to conclude that knowledge is incomplete. This chain is valid but analytic: it shows that if one defines open-endedness as the arrival of new predicates, then individual knowledge is incomplete. The paper should state this explicitly, because OEE2–OEE5 inherit the definitional character of OEE1 rather than gaining independent empirical or mathematical support.

minor comments (5)

[Definition 13] The definition text says "the quotient of the temporally adjacent contextual knowledge possible K_{i,t+1} with the current contextual knowledge possible K_{i,t+1}," but the displayed formula and the prose clearly require K_{i,t}, not K_{i,t+1}, in the second position.
[Equations (3.2), (6.3), and Proposition 8] There are several notation slips: δ_{i,t}(ξ) + δ_{it},(¬ξ) contains a stray comma in the second term; Proposition 8 refers to "Theorems 7 and 5" when only Theorem 5 exists; and the appendix refers to "SE5" where Section 2.1 uses "S5n." These should be corrected.
[Proposition 2(ii)] The statement "T_{i,t} = P_{i,t}" equates a theory with a set of nonlogical constants. Since a theory is a set of sentences and P_{i,t} is a set of predicates, this equality is a category error unless an unconventional identification is intended, which should be defined.
[Appendix, Proposition 6] The proof contains apparent typographical repetitions ("T_{i,t} ⊂ T_Ω, T_{i,t} ⊂ T_Ω") and does not actually prove the claimed generic inconsistency; it merely asserts that equality of theories "is not true under open-ended evolution," which is the conjecture.
[Section 4.6] The discussion of the "ragged partition" of system-level time is informal and does not engage with the formal definitions of T used elsewhere; it reads as a modeling suggestion rather than a result, and should be labeled as such.

Circularity Check

5 steps flagged · score 7.0 of 10

OEE2/OEE3 rest on the unproved Disjointness Conjecture used as a premise, while OEE1 and OEE5 restate the definition of open-endedness.

self definitional [Appendix 6.5, Proposition 5 (also Section 3.2)]
"Suppose A_{i,t}=∅. Then P_{i,t+1}=P_{i,t} by the definition of δ_{i,t}. But OEE systems are defined such that P_{i,t+1}\P_{i,t}≠∅. Therefore, A_{i,t}≠∅."

Definition 7 defines an OEE process by P_{i,t+1}\P_{i,t}≠∅, i.e., the appearance of new predicates is the definition of open-endedness. Proposition 5's proof uses exactly this defining condition to conclude A_{i,t}≠∅; it is the same fact restated in knowledge-space terminology. Lemma 1 and Theorem 2 then cite this nonemptiness to prove that individual knowledge is incomplete, so the incompleteness claim is a paraphrase of the definition rather than a derived theorem.
other [Section 3.2, Conjecture 1 (Disjointness)]
"Conjecture 1. (Disjointness) Under open-ended evolution, local knowledge is at least partially disjoint at any cross section of time and between cross sections. That is, K_{i,t}≠K_{j,t} in general for i,j∈N,t∈T."

The paper labels cross-agent disjointness as a conjecture and gives only a sketch. The sketch argues from Theorem 1, which states only that one agent's extension is essential relative to that agent's own earlier theory when new predicates appear; it says nothing about two agents' theories or knowledge sets converging to each other. The conjecture is therefore not derived; it is an additional assumption. The paper's central result OEE2 (common knowledge generally impossible) is then presented as following from this assumption, so the conclusion is imported as a premise.

3 more flagged steps

other [Appendix 6.5, Corollary 2 proof]
"Under open-ended evolution, T_{i,t}≠T_{j,t}≠T_Ω in general, which implies that P_{i,t}≠P_{j,t}≠P_Ω."

Corollary 2's proof begins with 'Under open-ended evolution, T_{i,t}≠T_{j,t}≠T_Ω in general,' which is exactly the cross-agent disjointness that Conjecture 1 was supposed to establish. No proof is supplied at this point; the previous material only established individual-level non-convergence (Theorem 1). Thus the impossibility of common knowledge is reduced to an unproved conjecture restated as a fact, and the proof does not go beyond its starting assumption.
other [Appendix 6.5, Proposition 6 proof]
"Suppose that, in general, T_{i,t}=T_{j,t}. But this would imply that K_{i,t}=K_{j,t} and Ω_{i,t}=Ω_{j,t} in general, which is not true under open-ended evolution."

Proposition 6 asserts that pairs of knowledge hierarchies are generally inconsistent and justifies it by 'which is not true under open-ended evolution' — i.e., it invokes the Disjointness Conjecture (K_{i,t}≠K_{j,t}, Ω_{i,t}≠Ω_{j,t}) as the reason. This is the same conjecture that Corollary 2 is meant to apply; the hierarchy-inconsistency result is therefore assumed, not proved.
self definitional [Section 4.5, Proposition 1]
"But by Proposition 5, A_{i,τ}≡K_{i,τ+1}\K_{i,τ}={ω′∉K_{i,τ}} is nonempty, meaning there exist states that lie in K_{i,τ+1} that i cannot integrate over. ... Therefore, they can never derive the right-hand side of the equation unless A_{i,τ} is empty—i.e., the system is closed."

The non-ergodicity proof uses Proposition 5's A_{i,τ} nonempty, which was derived directly from the defining condition P_{i,t+1}\P_{i,t}≠∅. The conclusion 'unless A_{i,τ} is empty—i.e., the system is closed' makes explicit that non-ergodicity is the definitional negation of closedness. The result restates open-endedness rather than deriving a separate fact about knowledge.

full rationale

The paper's formal apparatus is not circular through self-citation: the cited prior work by the authors (e.g., Koppl et al. 2023; Devereaux et al. 2024) is used for context and motivation, not as the proof of OEE1-5. However, the central derivation chain is substantially definitional. Open-endedness is defined as P_{i,t+1}\P_{i,t}≠∅ (Definition 7); Proposition 5 immediately converts this into nonempty adjacent knowledge possible, and Lemma 1 / Theorem 2 derive individual incompleteness from that same nonemptiness. The cross-agent disjointness needed for OEE2 and OEE3 is explicitly labeled Conjecture 1, and its proof sketch appeals to Theorem 1, which concerns only a single agent's theory extension, not convergence between agents. Proposition 6 and Corollary 2 then restate the conjecture as a premise ('T_{i,t}≠T_{j,t}≠T_Ω in general'), so the impossibility of common knowledge reduces to the very assumption meant to be established. Corollary 3 inherits this gap and, in addition, asserts Aumann-style 'agree to disagree' without defining posteriors or common priors. OEE5's non-ergodicity proof likewise uses the definitional nonemptiness of A_{i,τ}, making it a restatement of the defining predicate-growth condition. Thus OEE1, OEE2, OEE3, and OEE5 are not independently derived from first principles; they are either immediate unpackings of Definition 7 or consequences of an unproved conjecture reused as a premise. OEE4 is informal and does not introduce a separate circular step. Overall, the central negative results are substantially forced by the setup rather than established by independent argument.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The framework rests on a few definitional moves and two strong assumptions: PA interpretability of all theories and belief in local consistency and completeness. The headline results follow either from the definition of open-endedness or from the unproved Disjointness Conjecture. The ledger shows a high circularity burden, because the derivations largely unpack the authors' chosen definitions rather than confronting external benchmarks.

assumptions (6)

domain assumption A theory of everything TOmega exists for the universe, which is consistent and closed.
Section 4.1: 'Suppose we assume that the true universe Omega is consistent and closed and described by a theory-of-everything TOmega.' This is the benchmark against which observers are measured, but its existence is not derived.
domain assumption Every observer's language and theory interprets Peano arithmetic.
Section 6.1 states that L has enough algebra to express physical phenomena and 'interprets Peano arithmetic, as do all the major theories of physical and social systems.' This premise is needed to import Goedelian incompleteness into the framework and is asserted rather than argued.
ad hoc to paper Open-endedness is defined by the arrival of new nonlogical constants in every period, so P_{i,t+1}\P_{i,t} is nonempty.
Definition 7 equates open-endedness with nonempty growth in predicates. Proposition 5 then proves the adjacent knowledge possible is nonempty, and Theorem 2 derives incompleteness from it. The central results are encoded in this defining assumption.
ad hoc to paper Disjointness Conjecture: local knowledge of different observers is at least partially disjoint.
Section 3.2 states Conjecture 1 with only a proof sketch, then Corollary 2 in the Appendix uses 'Ti,t is not equal to Tj,t in general' as a fact. No independent proof is supplied, making this a load-bearing unproven premise.
domain assumption Each individual believes their own theory is consistent and complete.
Section 3.1: 'we assume that individual i believes their logical systems Ti,t to be consistent and complete.' This standard rationality assumption is used in Proposition 2 to ensure the decision procedure is defined everywhere, and it is nontrivial in an evolving system.
standard math Goedel's first incompleteness theorem and Tarski's undecidability results apply to each individual theory Ti,t.
Assumed as background mathematics in Section 4.3 and Theorem 3 of the Appendix. The application to social and economic theories depends on the earlier PA-interpretability axiom.

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Pith. "Pith review of The use of knowledge in open-ended systems." pith.science (2026). https://pith.science/paper/MFIZ7NZE

@misc{pith2026241200011,
  author       = {Pith},
  title        = {Pith review of: The use of knowledge in open-ended systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MFIZ7NZE}},
  note         = {Machine review of arXiv:2412.00011}
}

read the original abstract

Economists model knowledge use and acquisition as a cause-and-effect calculus associating observations made by a decision-maker about their world with possible underlying causes. Knowledge models are well-established for static contexts, but not for contexts of innovative and unbounded change. We develop a representation of knowledge use and acquisition in open-ended evolutionary systems and demonstrate its primary results, including that observers embedded in open-ended evolutionary systems can agree to disagree and that their ability to theorize about their systems is fundamentally local and constrained to their frame of reference what we call frame relativity. The results of our framework formalize local knowledge use, the many-selves interpretation of reasoning through time, and motivate the emergence of nonlogical modes of reasoning like institutional and aesthetic codes.

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