A Representation of Explicit Knowledge and Epistemic Indistinguishability in a Logic of Awareness

Satoshi Tojo; Yudai Kubono

arxiv: 2512.22477 · v1 · submitted 2025-12-27 · 💻 cs.LO

A Representation of Explicit Knowledge and Epistemic Indistinguishability in a Logic of Awareness

Yudai Kubono , Satoshi Tojo This is my paper

Pith reviewed 2026-05-16 19:46 UTC · model grok-4.3

classification 💻 cs.LO

keywords logic of awarenessexplicit knowledgeimplicit knowledgeepistemic logicindistinguishabilitysoundnesscompletenessmodal logic

0 comments

The pith

A refined logic of awareness defines explicit knowledge via awareness-dependent indistinguishability to block unwanted inferences from implicit knowledge.

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

The paper refines the logic of awareness introduced by Fagin and Halpern. Explicit knowledge is no longer simply implicit knowledge conjoined with awareness; instead, it requires that an agent cannot distinguish certain worlds once awareness is fixed. This change prevents Modus Ponens inside implicit knowledge from producing propositions that should not count as explicit knowledge. The authors illustrate the distinction with an elementary-geometry example in which students reach different conclusions depending on which learned facts they are aware of. They then introduce Awareness-Based Indistinguishability Logic (AIL), prove it strictly more expressive than the earlier framework while embedding that framework, and supply a sound and complete axiomatization.

Core claim

In the semantics of AIL an agent’s explicit knowledge of a formula holds only when the formula is true in all worlds indistinguishable under the awareness relation and the agent is aware of the formula; aware implicit knowledge is permitted to remain non-explicit. This relation is defined so that the indistinguishability classes shrink precisely when awareness increases, thereby avoiding the derivation of explicit knowledge from implicit knowledge alone via classical rules such as Modus Ponens.

What carries the argument

Awareness-based indistinguishability relation: a binary relation on possible worlds whose equivalence classes are determined by the agent’s current awareness set, used to interpret explicit knowledge separately from implicit knowledge.

If this is right

AIL can express distinctions between explicit and aware-implicit knowledge that the earlier logic collapses.
The Fagin-Halpern logic is faithfully embeddable inside AIL, so all prior results remain available as a special case.
A sound and complete axiomatic system exists for AIL, enabling formal verification of epistemic statements.
The geometry example demonstrates that different awareness sets produce different explicit-knowledge conclusions even when implicit knowledge is identical.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same indistinguishability construction could be used to model agents that acquire new explicit knowledge precisely when they become aware of a bridging lemma.
If awareness sets are allowed to change over time, AIL could serve as the static fragment of a dynamic epistemic logic for learning.
The framework supplies a natural setting for comparing the explicit knowledge of human reasoners who share the same implicit background but differ in noticed facts.

Load-bearing premise

The proposed awareness-dependent indistinguishability relation correctly separates explicit from merely aware implicit knowledge without letting classical rules generate new explicit propositions that the intended distinction rules out.

What would settle it

A concrete counter-example in which the geometry scenario yields an explicit-knowledge proposition under AIL that the original Fagin-Halpern definition also counts as explicit, or vice versa, would falsify the claimed improvement.

Figures

Figures reproduced from arXiv: 2512.22477 by Satoshi Tojo, Yudai Kubono.

**Figure 1.** Figure 1: An elementary geometry problem. Implicit Knowledge This comprises what an agent has independently of the agent’s current awareness. Furthermore, the knowledge closes under logical consequences and includes all tautologies, thus constituting the ideal knowledge of a logically omniscient agent. Explicit Knowledge This comprises what an agent has in the agent’s awareness and hence is available for reasoning o… view at source ↗

**Figure 2.** Figure 2: A proof of Figure 1. Note that the given hypotheses or derived facts are shown in the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: A possible world model where a set of possible worlds is [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: A possible world model where a set of possible worlds is [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: Two possible world models. Epistemic model with awareness [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: Partitions of a set W consisting of eight worlds by IK-accessibility, A-equivalence, and EK-accessibility relations. Given [w1]∼i and [w1]≈i (see (1) and (2)), w2 in [w1]≈i belongs to the partition by (∼i ◦ ≈i) +. Then, w3 is in this connected partition as w3 ∼i w2. Transitively, w4 is also in it as w3 ≈i w4. The final partition becomes as (3). is sufficient to prove, for every w, v ∈ W, if (w, v) ∈ (∼i ◦ … view at source ↗

**Figure 7.** Figure 7: An epistemic model with awareness for Example 4. For readability, we omit the [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗

**Figure 8.** Figure 8: Two epistemic models with awareness for a single agent [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗

**Figure 9.** Figure 9: Possible world models with awareness for a single agent, where a solid line represents [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗

read the original abstract

The logic of awareness, first proposed by Fagin and Halpern, addressed the problem of logical omniscience by introducing the notion of awareness and distinguishing explicit knowledge from implicit knowledge. In their framework, explicit knowledge was defined as the conjunction of implicit knowledge and awareness, each of which was represented by modal operators. Their definition, however, may derive undesirable propositions that cannot be considered explicit knowledge when Modus Ponens is applied within implicit knowledge. Hence, focusing on indistinguishability among possible worlds, dependent on awareness, we refine the definition of explicit knowledge. In our semantics, we require that the aware implicit knowledge is not necessarily explicit knowledge, though explicit knowledge must be aware as well as implicit. We employ an example of elementary geometry, where different students may or may not reach the final answer, depending on whether they are aware of learned mathematical facts. Thereafter, we formally present the syntax and the semantics of our language, named Awareness-Based Indistinguishability Logic ($\mathrm{AIL}$). We prove that $\mathrm{AIL}$ has more expressive power than the logic of Fagin and Halpern, and show that the latter is embeddable in $\mathrm{AIL}$. Furthermore, we provide an axiomatic system of $\mathrm{AIL}$ and prove its soundness and completeness.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper fixes the MP-derived explicit knowledge problem in Fagin-Halpern by making indistinguishability depend on awareness, then shows AIL is strictly more expressive, embeds the old logic, and supplies sound complete axioms.

read the letter

The main point is a targeted fix to how explicit knowledge is defined. In the original Fagin-Halpern setup, explicit knowledge was implicit knowledge conjoined with awareness, but that allowed modus ponens inside implicit knowledge to produce explicit knowledge that agents should not count as having. Kubono and Tojo make the indistinguishability relation itself depend on what the agent is aware of, so that aware implicit knowledge is no longer automatically explicit while explicit knowledge still requires both awareness and implicit knowledge. They call the resulting system AIL. They prove it is strictly more expressive than the Fagin-Halpern logic and that the older logic embeds into AIL. They also give an axiomatic system and claim soundness and completeness. The elementary geometry example shows the intended distinction in a concrete way: students who have learned the same facts implicitly reach different conclusions depending on which facts they are aware of. That illustration is clear and stays within the motivation. The central semantic move looks like a reasonable way to block the unwanted closure without adding extra operators. The expressivity and embedding results are the clearest new technical content. The axiomatization follows standard modal techniques once the semantics is fixed, so the proofs are probably routine once the relation is defined. The main soft spot is that the paper treats the new indistinguishability relation as correctly capturing the distinction without introducing fresh unwanted propositions; the geometry case supports the motivation but does not exhaustively test side effects. If the full proofs hold up under checking, the argument stands. This is for people working on epistemic logics with bounded awareness, especially those modeling agents that do not close under all logical consequences. A reader who already knows the Fagin-Halpern framework will see the refinement quickly. It is focused enough and the metatheoretic claims are precise enough that it deserves a serious referee rather than a desk rejection.

Referee Report

1 major / 3 minor

Summary. The manuscript introduces Awareness-Based Indistinguishability Logic (AIL) as a refinement of Fagin and Halpern's logic of awareness. It redefines explicit knowledge via an awareness-dependent indistinguishability relation on possible worlds, ensuring explicit knowledge requires both implicit knowledge and awareness while allowing aware implicit knowledge to fall short of explicit knowledge. This blocks certain undesirable propositions derivable by modus ponens on implicit knowledge alone. The paper proves AIL is strictly more expressive than Fagin-Halpern logic, exhibits an embedding of the latter into AIL, and supplies an axiomatic system together with soundness and completeness proofs. The distinction is motivated by an elementary geometry example involving students' awareness of learned facts.

Significance. If the metatheoretic claims hold, the work supplies a semantically grounded improvement to the modeling of explicit knowledge that avoids a known shortcoming of the original Fagin-Halpern definition. The strict expressivity increase combined with the embedding result positions AIL as a conservative yet stronger framework. The provision of a sound and complete axiomatization is a concrete strength that supports further technical development and potential applications in multi-agent epistemic reasoning.

major comments (1)

§5 (Axiomatization and metatheory): the manuscript asserts soundness and completeness for the proposed axiomatic system, yet the detailed derivations, key lemmas, and canonical-model construction are not exhibited. Without these, it is impossible to verify that the axioms fully capture the awareness-dependent indistinguishability relation and that no unintended validities are introduced.

minor comments (3)

Introduction and §2: the geometry example illustrates the intended distinction but remains informal; translating the scenario into explicit AIL formulas (e.g., showing a concrete instance where aware implicit knowledge fails to be explicit) would make the motivation sharper and easier to check against the semantics.
§3 (Semantics): the definition of the new indistinguishability relation R_a^w should be accompanied by a short verification that it preserves the frame properties needed for the modal fragment (reflexivity, etc.) so that standard modal validity results continue to hold where expected.
Notation throughout: ensure uniform typesetting of the explicit-knowledge operator and the awareness-dependent accessibility relation across all sections and figures.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation for minor revision. The single major comment is addressed point-by-point below.

read point-by-point responses

Referee: §5 (Axiomatization and metatheory): the manuscript asserts soundness and completeness for the proposed axiomatic system, yet the detailed derivations, key lemmas, and canonical-model construction are not exhibited. Without these, it is impossible to verify that the axioms fully capture the awareness-dependent indistinguishability relation and that no unintended validities are introduced.

Authors: We agree that the detailed derivations, key lemmas, and canonical-model construction must be exhibited for the metatheoretic claims to be verifiable. The current manuscript presents the axiomatization and states the soundness and completeness results but condenses the supporting arguments. In the revised version we will expand §5 to include the full set of key lemmas, the explicit construction of the canonical model, and the step-by-step verification that the axioms characterize precisely the awareness-dependent indistinguishability relation without introducing extraneous validities. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines a new semantics for AIL that refines explicit knowledge via an awareness-dependent indistinguishability relation, proves greater expressive power than Fagin-Halpern logic plus embeddability, and supplies a sound and complete axiomatization. All steps use standard modal-logic semantic arguments and metatheoretic techniques on the newly introduced structures; none reduce by construction to prior inputs, fitted parameters, or self-citations. The geometry example appears only as informal motivation and does not enter any formal derivation or equation. No load-bearing self-citation chains or ansatz smuggling are present.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard modal logic background and the prior Fagin-Halpern framework; no new free parameters or invented entities are introduced.

axioms (1)

standard math Standard Kripke semantics for modal operators of knowledge and awareness
Invoked to define the new indistinguishability relation.

pith-pipeline@v0.9.0 · 5530 in / 1208 out tokens · 21503 ms · 2026-05-16T19:46:26.301583+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that AIL has more expressive power than the logic of Fagin and Halpern... axiomatic system of AIL and prove its soundness and completeness.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

E_i φ ↔ A_i φ ∧ [◦+]i φ (EK-accessibility via (∼_i ◦ ≈_i)+)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

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Awareness and forgetting of facts and agents

Hans van Ditmarsch and Tim French. Awareness and forgetting of facts and agents. In 2009 IEEE/WIC/ACM International Joint Conference on Web Intelligence and Intelligent Agent Technology, pages 478–483, 2009

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Becoming aware of propositional variables

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Vel´ azquez-Quesada, and Y` ı N

Hans van Ditmarsch, Tim French, Fernando R. Vel´ azquez-Quesada, and Y` ı N. W´ ang. Im- plicit, explicit and speculative knowledge.Artificial Intelligence, 256:35–67, 2018

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[1] [1]

A logic for reasoning about knowledge of unaware- ness.Journal of Logic, Language and Information, 23(2):197–217, 2014

Thomas ˚Agotnes and Natasha Alechina. A logic for reasoning about knowledge of unaware- ness.Journal of Logic, Language and Information, 23(2):197–217, 2014

work page 2014

[2] [2]

A logical theory of be- lief dynamics for resource-bounded agents

Philippe Balbiani, David Fern´ andez-Duque, and Emiliano Lorini. A logical theory of be- lief dynamics for resource-bounded agents. In15th International Joint Conference on Au- tonomous Agents and Multiagent Systems (AAMAS 2016), pages 644–652, 2016

work page 2016

[3] [3]

Rendsvig

Gaia Belardinelli and Rasmus K. Rendsvig. Awareness logic: Kripke lattices as a middle ground between syntactic and semantic models.J. Log. Comput., 33(6):1186–1215, 2023

work page 2023

[4] [4]

Schipper

Gaia Belardinelli and Burkhard C. Schipper. Implicit knowledge in unawareness structures. Synthese, 204(5):141, 2024

work page 2024

[5] [5]

Clarendon Press, 2006

Michael Bergmann.Justification without awareness: A defense of epistemic externalism. Clarendon Press, 2006

work page 2006

[6] [6]

Chellas.Modal logic: an introduction

Brian F. Chellas.Modal logic: an introduction. Cambridge University Press, 1980

work page 1980

[7] [7]

Ronald Fagin and Joseph Y. Halpern. Belief, awareness, and limited reasoning.Artificial Intelligence, 34:39–76, 1988

work page 1988

[8] [8]

Halpern, Yoram Moses, and Moshe Y

Ronald Fagin, Joseph Y. Halpern, Yoram Moses, and Moshe Y. Vardi.Reasoning About Knowledge. MIT Press, 1995

work page 1995

[9] [9]

Springer, 2021

Claudia Fern´ andez-Fern´ andez.Awareness in Logic and Epistemology: A Conceptual Schema and Logical Study of the Underlying Main Epistemic Concepts. Springer, 2021. 26

work page 2021

[10] [10]

Vel´ azquez-Quesada

Claudia Fern´ andez-Fern´ andez and Fernando R. Vel´ azquez-Quesada. Awareness of and awareness that: their combination and dynamics.Logic Journal of the IGPL, 29(4):601–626, 2021

work page 2021

[11] [11]

A note on brute vs

Davide Grossi. A note on brute vs. institutional facts. InNormative Multi-Agent Systems, 15.03. - 20.03.2009, volume 09121 ofDagstuhl Seminar Proceedings, 2009

work page 2009

[12] [12]

The ceteris paribus structure of logics of game forms.Journal of Artificial Intelligence Research, 53:91–126, 2015

Davide Grossi, Emiliano Lorini, and Fran¸ cois Schwarzentruber. The ceteris paribus structure of logics of game forms.Journal of Artificial Intelligence Research, 53:91–126, 2015

work page 2015

[13] [13]

Vel´ azquez-Quesada

Davide Grossi and Fernando R. Vel´ azquez-Quesada. Syntactic awareness in logical dynamics. Synthese, 192(12):4071–4105, 2015

work page 2015

[14] [14]

Joseph Y. Halpern. Alternative semantics for unawareness.Games and Economic Behavior, 37(2):321–339, 2001

work page 2001

[15] [15]

Halpern and Leandro C

Joseph Y. Halpern and Leandro C. Rˆ ego. Interactive unawareness revisited.Games and Economic Behavior, 62(1):232–262, 2008

work page 2008

[16] [16]

Schipper

Aviad Heifetz, Martin Meier, and Burkhard C. Schipper. Interactive unawareness.Journal of Economic Theory, 130(1):78–94, 2006

work page 2006

[17] [17]

Schipper

Aviad Heifetz, Martin Meier, and Burkhard C. Schipper. A canonical model for interactive unawareness.Games and Economic Behavior, 62:304–324, 2008

work page 2008

[18] [18]

Schipper

Aviad Heifetz, Martin Meier, and Burkhard C. Schipper. Dynamic unawareness and ratio- nalizable behavior.Games and Economic Behavior, 81:50–68, 2013

work page 2013

[19] [19]

Schipper

Aviad Heifetz, Martin Meier, and Burkhard C. Schipper. Unawareness, beliefs, and specu- lative trade.Games and Economic Behavior, 77(1):100–121, 2013

work page 2013

[20] [20]

Logic of awareness in agent’s reason- ing

Yudai Kubono, Teeradaj Racharak, and Satoshi Tojo. Logic of awareness in agent’s reason- ing. InProceedings of the 15th International Conference on Agents and Artificial Intelligence - Volume 1: ICAART, pages 207–216, 2023. [Errata arXiv:2309.09214]

work page arXiv 2023

[21] [21]

Grounding awareness on belief bases

Emiliano Lorini and Pengfei Song. Grounding awareness on belief bases. InInternational Workshop on Dynamic Logic, pages 170–186. Springer, 2020

work page 2020

[22] [22]

Schipper

Burkhard C. Schipper. Awareness. In Hans van Ditmarsch, Wiebe van der Hoek, Joseph Y. Halpern, and Barteld Kooi, editors,Handbook of epistemic logic. College Publications, 2015

work page 2015

[23] [23]

Course Technology, second edition, 2006

Michael Sipser.Introduction to the Theory of Computation. Course Technology, second edition, 2006

work page 2006

[24] [24]

Globally rigid awareness logic with static abstrac- tion operators

Kosuke Udatsu and Katsuhiko Sano. Globally rigid awareness logic with static abstrac- tion operators. InInternational Workshop on Logical Aspects in Multi-Agent Systems and Strategic Reasoning (LAMAS&SR2024), 2024

work page 2024

[25] [25]

Cambridge University Press, 2011

Johan van Benthem.Logical dynamics of information and interaction. Cambridge University Press, 2011

work page 2011

[26] [26]

Vel´ azquez-Quesada

Johan van Benthem and Fernando R. Vel´ azquez-Quesada. The dynamics of awareness. Synthese, 177(1):5–27, 2010. 27

work page 2010

[27] [27]

Awareness and forgetting of facts and agents

Hans van Ditmarsch and Tim French. Awareness and forgetting of facts and agents. In 2009 IEEE/WIC/ACM International Joint Conference on Web Intelligence and Intelligent Agent Technology, pages 478–483, 2009

work page 2009

[28] [28]

Becoming aware of propositional variables

Hans van Ditmarsch and Tim French. Becoming aware of propositional variables. InLogic and Its Applications, pages 204–218, 2011

work page 2011

[29] [29]

Vel´ azquez-Quesada, and Y` ı N

Hans van Ditmarsch, Tim French, Fernando R. Vel´ azquez-Quesada, and Y` ı N. W´ ang. Im- plicit, explicit and speculative knowledge.Artificial Intelligence, 256:35–67, 2018

work page 2018

[30] [30]

Springer Science & Business Media, 2007

Hans van Ditmarsch, Wiebe van der Hoek, and Barteld Kooi.Dynamic Epistemic Logic. Springer Science & Business Media, 2007

work page 2007

[31] [31]

Inexact knowledge.Mind, 101(402):217–242, 1992

Timothy Williamson. Inexact knowledge.Mind, 101(402):217–242, 1992. 28

work page 1992