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arxiv: 2606.31888 · v1 · pith:YTZDFW3Tnew · submitted 2026-06-30 · 💻 cs.LO

Non-classical Topological Evidence Logic

Igor Sedl\'ar (Institute of Computer Science , Czech Academy of Sciences) This is my paper

Pith reviewed 2026-07-01 02:28 UTC · model grok-4.3

classification 💻 cs.LO
keywords topological evidence logicrelevant logicintuitionistic logicepistemic logicsoundness and completenessglobal modalitydense open setsinterior operator
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The pith

Topological Evidence Logic remains sound and complete when the base logic shifts from classical to intuitionistic or relevant.

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

The paper establishes that coherent epistemic justification, defined as entailment by a dense open set in a topological space, can be expressed without relying on classical propositional logic. It constructs an intuitionistic version by extending a recent modal framework with a global modality. For relevant logic it augments the weak relevant modal logic BS4 with an interior-of-complement operator, then proves soundness and completeness for the resulting system. A sympathetic reader would care because ordinary agents often reason in non-classical ways, and the topological model of justification continues to work under those changes.

Core claim

An extension of the intuitionistic modal framework with a global modality expresses coherent justification intuitionistically, while an interior-of-complement operator added to the weak relevant modal logic BS4 yields a sound and complete relevant version of Topological Evidence Logic.

What carries the argument

Topological semantics in which a hypothesis is coherently justified precisely when it is entailed by a dense open set, lifted to non-classical bases via a global modality and, for relevance, an interior-of-complement operator.

If this is right

  • Coherent justification receives a sound and complete intuitionistic formalization.
  • Relevant Topological Evidence Logic based on BS4 is sound and complete.
  • The topological approach to epistemic justification does not depend on classical propositional logic.
  • A global modality suffices to express the topological condition in both non-classical settings.

Where Pith is reading between the lines

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

  • The same technique might allow coherent-justification semantics inside other substructural logics that already possess suitable modalities.
  • If agents reason relevantly, the interior-of-complement operator supplies a minimal syntactic addition that restores the topological reading without restoring classical explosion.
  • The result suggests that topological models of evidence can be ported to reasoning systems that reject contraction or weakening.

Load-bearing premise

The topological definition of coherent justification via dense open sets can be preserved when the underlying logic is made intuitionistic or relevant by adding the required operators.

What would settle it

A countermodel in the relevant semantics that satisfies all axioms and rules of the BS4-based system yet fails to validate a formula that should be valid under the topological reading.

Figures

Figures reproduced from arXiv: 2606.31888 by Czech Academy of Sciences), Igor Sedl\'ar (Institute of Computer Science.

Figure 1
Figure 1. Figure 1: Modal axioms and rules of the system iS4A, where ♡ ∈ {□,A}. 3 Axiomatisation of Intuitionistic TEL To the best of our knowledge, there is no explicit axiomatisation of intuitionistic modal logic with A in the literature. As discussed in [35, p. 1304], for example, intuitionistic propositional logic with A can be axiomatised using Ono’s system L4 from [22]. Here, we extend this axiomatisation to fit the lan… view at source ↗
Figure 2
Figure 2. Figure 2: A model showing that interior of complement is not expressible in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A pair of models showing that support by a dense open set is not expressible in [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A model showing that intuitionistic implication is not expressible in [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Provability in BS4A,c is defined in the standard way. We write ⊢ ϕ if ϕ is provable in BS4A,c. Axioms (13–18) for A say that A is a relevant S5-style modality; see [33]. (19–21) say that □cϕ expresses the largest open proposition disjoint from the proposition expressed by ϕ. Axioms (22–29) ϕ → ϕ (1) (ϕ ∧ψ) → ϕ, (ϕ ∧ψ) → ψ (2) ϕ → (ϕ ∨ψ), ψ → (ϕ ∨ψ) (3) ((ϕ → ψ)∧(ϕ → χ)) → (ϕ → (ψ ∧χ)) (4) ((ϕ → χ)∧(ψ → χ))… view at source ↗
Figure 5
Figure 5. Figure 5: A Hilbert-style system B for the basic relevant logic B [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The extra axioms and rules of BS4A,c, to be added to B, where ♡ ∈ {□,A}. impose uniformity of formulas with the global modality under ∗ and R and within open sets; note that, in the words of Standefer and French [35], a certain amount of classicality is thereby imposed. These axioms are crucial for showing that the anchored canonical model is an RM up-model, as well as for proving the Truth Lemma for the m… view at source ↗
read the original abstract

Topological Evidence Logic (TEL) is a recent approach to epistemic logic that uses topological tools to model coherent epistemic justification. Specifically, a hypothesis is coherently justified if and only if it is entailed by a dense open set. In its simplest form, TEL can be formulated as an extension of S4 with a global modality. All currently studied forms of TEL are based on classical propositional logic, which has been heavily criticised for misrepresenting the way in which ordinary agents reason. In this article, we show that the TEL approach is robust under modifications to the propositional base. We show that an extension of the intuitionistic modal framework recently introduced by de Groot and Shillito, incorporating a global modality, enables coherent justification to be expressed in an intuitionistic setting. Furthermore, we adapt the recent work of Standefer et al., which extends relevant logic with a global modality, to show that coherent justification can be expressed in a relevant setting if an interior-of-complement operator is added to the language. Our main technical result is a soundness and completeness theorem for relevant TEL based on the weak relevant modal logic BS4.

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.

Referee Report

0 major / 2 minor

Summary. The paper extends Topological Evidence Logic (TEL) beyond classical propositional logic to intuitionistic and relevant settings. It incorporates a global modality into the intuitionistic modal framework of de Groot and Shillito to express coherent justification (entailment by a dense open set) in an intuitionistic context, and adapts the relevant-logic global-modality construction of Standefer et al. by adding an interior-of-complement operator, yielding a soundness-and-completeness theorem for relevant TEL based on the weak relevant modal logic BS4.

Significance. If the soundness and completeness results hold, the work demonstrates that the TEL approach is robust under changes to the propositional base, allowing coherent epistemic justification to be modeled in logics that avoid well-known limitations of classical propositional logic for representing ordinary reasoning. The main technical contribution—the completeness theorem for relevant TEL—is an independent extension of the cited frameworks rather than a restatement, and the absence of ad-hoc axioms or free parameters in the construction is a strength.

minor comments (2)
  1. [Abstract] Abstract: the claim that 'the topological semantics lifts while preserving the key properties needed for the completeness proof' would benefit from a one-sentence pointer to the specific lemma or proposition that verifies preservation of the relevant frame conditions under the interior-of-complement operator.
  2. [Introduction] The manuscript would be easier to follow if the introduction included a short paragraph recalling the key semantic clauses of the de Groot–Shillito intuitionistic framework and the Standefer et al. relevant global-modality construction before describing the modifications.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript, positive assessment of its significance, and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives its main soundness-and-completeness result for relevant TEL by adapting the external intuitionistic modal framework of de Groot and Shillito (with global modality) and the relevant-logic global-modality construction of Standefer et al., then adjoining an interior-of-complement operator while preserving the required frame properties. These supporting results are cited from independent prior work by different authors; the new theorem is obtained by standard adaptation rather than by redefining any operator or parameter in terms of the target theorem itself. No self-citation is load-bearing, no ansatz is smuggled, and no prediction reduces to a fitted input by construction. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new entities; the result is presented as a technical adaptation of existing modal frameworks.

pith-pipeline@v0.9.1-grok · 5726 in / 997 out tokens · 56097 ms · 2026-07-01T02:28:00.555854+00:00 · methodology

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Works this paper leans on

40 extracted references · 29 canonical work pages

  1. [1]

    Annals of Pure and Applied Logic 51(1–2), pp

    Samson Abramsky (1991): Domain theory in logical form. Annals of Pure and Applied Logic 51(1–2), pp. 1–77, doi:10.1016/0168-0072(91)90065-t

    work page doi:10.1016/0168-0072(91)90065-t 1991

  2. [2]

    Journal of Logic and Computation 13(6), pp

    Marco Aiello, Johan van Benthem & Guram Bezhanishvili (2003): Reasoning About Space: The Modal Way. Journal of Logic and Computation 13(6), pp. 889–920, doi:10.1093/logcom/13.6.889. I. Sedlár 709

    work page doi:10.1093/logcom/13.6.889 2003

  3. [3]

    Artemov & T

    Sergei Artemov & Tudor Protopopescu (2016): Intuitionistic Epistemic Logic . The Review of Symbolic Logic 9(2), pp. 266–298, doi:10.1017/s1755020315000374

    work page doi:10.1017/s1755020315000374 2016

  4. [4]

    Artificial Intelligence 349, p

    Alexandru Baltag, Nick Bezhanishvili & David Fernández-Duque (2025): The topology of surprise. Artificial Intelligence 349, p. 104423, doi:10.1016/j.artint.2025.104423

    work page doi:10.1016/j.artint.2025.104423 2025

  5. [5]

    In: Language, Logic, and Computation (TbiLLC 2019), Springer International Publish- ing, pp

    Alexandru Baltag, Nick Bezhanishvili & Saúl Fernández González (2022): Topological Evidence Logics: Multi-agent Setting. In: Language, Logic, and Computation (TbiLLC 2019), Springer International Publish- ing, pp. 237–257, doi:10.1007/978-3-030-98479-3_12

    work page doi:10.1007/978-3-030-98479-3_12 2022

  6. [6]

    Synthese 200(6), doi:10.1007/s11229-022-03967-6

    Alexandru Baltag, Nick Bezhanishvili, Aybüke Özgün & Sonja Smets (2022): Justified belief, knowledge, and the topology of evidence. Synthese 200(6), doi:10.1007/s11229-022-03967-6

    work page doi:10.1007/s11229-022-03967-6 2022

  7. [7]

    In: Logic, Language, Information, and Computation , Springer Berlin Heidelberg, pp

    Alexandru Baltag, Nick Bezhanishvili, Aybüke Özgün & Sonja Smets (2016): Justified Belief and the Topol- ogy of Evidence . In: Logic, Language, Information, and Computation , Springer Berlin Heidelberg, pp. 83–103, doi:10.1007/978-3-662-52921-8_6

    work page doi:10.1007/978-3-662-52921-8_6 2016

  8. [8]

    In: Eu- ropean Conference on Computer Vision (2020),https://doi.org/10.1007/978- 3-030-58452-8_241

    Alexandru Baltag, Nick Bezhanishvili, Aybüke Özgün & Sonja Smets (2017):The Topology of Full and Weak Belief. In: Logic, Language, and Computation, Springer Berlin Heidelberg, pp. 205–228, doi:10.1007/978- 3-662-54332-0_12

    work page doi:10.1007/978- 2017

  9. [9]

    Technical Report, doi:10.48550/arXiv.2509.00184

    Alexandru Baltag, Malvin Gattinger & Djanira Gomes (2025): Virtual Group Knowledge and Group Belief in Topological Evidence Models (Extended Version). Technical Report, doi:10.48550/arXiv.2509.00184. To appear in the proceedings of DaLí 2025

    work page doi:10.48550/arxiv.2509.00184 2025

  10. [10]

    Journal of Logic and Computation 26(6), pp

    Marta Bílková, Ondrej Majer & Michal Peliš (2016): Epistemic logics for sceptical agents. Journal of Logic and Computation 26(6), pp. 1815–1841, doi:10.1093/logcom/exv009

    work page doi:10.1093/logcom/exv009 2016

  11. [11]

    The Review of Symbolic Logic 13(4), pp

    Adam Bjorndahl & Aybüke Özgün (2019): Logic and Topology for Knowledge, Knowability, and Belief. The Review of Symbolic Logic 13(4), pp. 748–775, doi:10.1017/s1755020319000509

    work page doi:10.1017/s1755020319000509 2019

  12. [12]

    Cambridge University Press, 2001

    Patrick Blackburn, Maarten de Rijke & Yde Venema (2001): Modal Logic. Cambridge University Press, Cambridge, doi:10.1017/cbo9781107050884

    work page doi:10.1017/cbo9781107050884 2001

  13. [13]

    Journal of Logic and Computation 35(7), doi:10.1093/logcom/exae030

    Jim de Groot & Ian Shillito (2025): Intuitionistic S4 as a logic of topological spaces . Journal of Logic and Computation 35(7), doi:10.1093/logcom/exae030

    work page doi:10.1093/logcom/exae030 2025

  14. [14]

    Michael Dunn (1993): Star and Perp: Two Treatments of Negation

    J. Michael Dunn (1993): Star and Perp: Two Treatments of Negation . Philosophical Perspectives 7, pp. 331–357, doi:10.2307/2214128

    work page doi:10.2307/2214128 1993

  15. [15]

    Electronic Notes in Theo- retical Computer Science 87, pp

    Martín Escardó (2004): Synthetic Topology of Data Types and Classical Spaces. Electronic Notes in Theo- retical Computer Science 87, pp. 21–156, doi:10.1016/j.entcs.2004.09.017

    work page doi:10.1016/j.entcs.2004.09.017 2004

  16. [16]

    Saúl Fernández González (2018): Generic Models for Topological Evidence Logics . MSc. Thesis, ILLC, University of Amsterdam, Amsterdam

    2018

  17. [17]

    Inda- gationes mathematicae 26(5), pp

    Andrzej Grzegorczyk (1964): A philosophically plausible formal interpretation of intuitionistic logic. Inda- gationes mathematicae 26(5), pp. 596–601, doi:10.1016/s1385-7258(64)50066-9

    work page doi:10.1016/s1385-7258(64)50066-9 1964

  18. [18]

    Kelly (1996): The Logic of Reliable Inquiry

    Kevin T. Kelly (1996): The Logic of Reliable Inquiry. Oxford University Press

    1996

  19. [19]

    J. C. C. McKinsey (1941): A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology. The Journal of Symbolic Logic 6(4), pp. 117–134, doi:10.2307/2267105

    work page doi:10.2307/2267105 1941

  20. [20]

    J. C. C. McKinsey & Alfred Tarski (1944): The Algebra of Topology. The Annals of Mathematics 45(1), p. 141, doi:10.2307/1969080

    work page doi:10.2307/1969080 1944

  21. [21]

    Munkres (2000): Topology, second edition

    James R. Munkres (2000): Topology, second edition. Prentice Hall

    2000

  22. [22]

    Publications of the Research Institute for Math- ematical Sciences 13(3), pp

    Hiroakira Ono (1977): On some intuitionistic modal logics. Publications of the Research Institute for Math- ematical Sciences 13(3), pp. 687–722, doi:10.2977/prims/1195189604

    work page doi:10.2977/prims/1195189604 1977

  23. [23]

    PhD thesis, ILLC, Univer- sity of Amsterdam

    Aybüke Özgün (2017): Evidence in Epistemic Logic: A Topological Perspective. PhD thesis, ILLC, Univer- sity of Amsterdam. Available at https://eprints.illc.uva.nl/id/eprint/2147/. DS-2017-07

    2017

  24. [24]

    Routledge, London, doi:10.4324/9780203016244

    Greg Restall (2000): An Introduction to Substructural Logics . Routledge, London, doi:10.4324/9780203016244. 710 Non-classical TEL

    work page doi:10.4324/9780203016244 2000

  25. [25]

    Journal of Cognitive Science 6, pp

    Greg Restall (2005): Logics, Situations and Channels. Journal of Cognitive Science 6, pp. 125–150

    2005

  26. [26]

    Meyer (1973): Semantics of entailment

    Richard Routley & Robert K. Meyer (1973): Semantics of entailment . In Hugues Leblanc, editor: Truth, Syntax and Modality, North Holland, Amsterdam, pp. 194–243

    1973

  27. [27]

    Meyer & Ross T

    Richard Routley, Val Plumwood, Robert K. Meyer & Ross T. Brady (1982): Relevant Logics and Their Rivals. 1, Ridgeview

    1982

  28. [28]

    Journal of Applied Non-Classical Logics 25(3), pp

    Igor Sedlár (2015): Substructural epistemic logics . Journal of Applied Non-Classical Logics 25(3), pp. 256–285, doi:10.1080/11663081.2015.1094313

    work page doi:10.1080/11663081.2015.1094313 2015

  29. [29]

    In David Fernández-Duque, Alessandra Palmigiano & Sophie Pinchinat, editors: Proc

    Igor Sedlár & Pietro Vigiani (2022): Relevant Reasoners in a Classical World. In David Fernández-Duque, Alessandra Palmigiano & Sophie Pinchinat, editors: Proc. 14th Int. Conference on Advances in Modal Logic (AiML 2022), College Publications, London, pp. 697–719. Available at https://arxiv.org/abs/2206. 03109

    2022

  30. [30]

    Journal of Philosophical Logic 53(5), pp

    Igor Sedlár & Pietro Vigiani (2024): Epistemic Logics for Relevant Reasoners . Journal of Philosophical Logic 53(5), pp. 1383–1411, doi:10.1007/s10992-024-09770-7

    work page doi:10.1007/s10992-024-09770-7 2024

  31. [31]

    M. B. Smyth (1983): Power domains and predicate transformers: A topological view . In J. Diaz, editor: Automata, Languages and Programming (ICALP 1983) , Springer-Verlag, pp. 662–675, doi:10.1007/bfb0036946

    work page doi:10.1007/bfb0036946 1983

  32. [32]

    M. B. Smyth (1992): Topology. In S. Abramsky, Dov M. Gabbay & T. S. E. Maibaum, edi- tors: Handbook of Logic in Computer Science , 1, Oxford University Press, Oxford, pp. 641–761, doi:10.1093/oso/9780198537359.003.0005

    work page doi:10.1093/oso/9780198537359.003.0005 1992

  33. [33]

    Logic and Logical Philosophy 32(1), pp

    Shawn Standefer (2022): Varieties of Relevant S5 . Logic and Logical Philosophy 32(1), pp. 53–80, doi:10.12775/llp.2022.011

    work page doi:10.12775/llp.2022.011 2022

  34. [34]

    Implication, Modality, Quantification

    Shawn Standefer (2026): Relevant Logics. Implication, Modality, Quantification . Cambridge University Press

    2026

  35. [35]

    Journal of Applied Logics – IfCoLog Journal 12(5), pp

    Shawn Standefer & Rohan French (2025): Universal Necessity and Deep Classicality . Journal of Applied Logics – IfCoLog Journal 12(5), pp. 1303–1318

    2025

  36. [36]

    Journal of Philosophical Logic, doi:10.1007/s10992-025-09791-w

    Shawn Standefer & Edwin Mares (2025): Symmetry and Completeness in Relevant Epistemic Logic. Journal of Philosophical Logic, doi:10.1007/s10992-025-09791-w

    work page doi:10.1007/s10992-025-09791-w 2025

  37. [37]

    Asian Journal of Philosophy 2(2), doi:10.1007/s44204-023-00065-3

    Shawn Standefer, Ted Shear & Rohan French (2023): Getting some (non-classical) closure with justification logic. Asian Journal of Philosophy 2(2), doi:10.1007/s44204-023-00065-3

    work page doi:10.1007/s44204-023-00065-3 2023

  38. [38]

    Fundamenta Mathematicae 38(1), pp

    Alfred Tarski (1938): Der Aussagenkalkül und die Topologie . Fundamenta Mathematicae 38(1), pp. 103– 134, doi:10.4064/fm-31-1-103-134

    work page doi:10.4064/fm-31-1-103-134 1938

  39. [39]

    Cambridge University Press

    Steven Vickers (1989): Topology via Logic. Cambridge University Press

    1989

  40. [40]

    Aybüke Özgün, Sonja Smets & Teodor-Stefan Zotescu (2025): Evidence Diffusion in Social Networks: a Topological Perspective. In V . Goranko, C. Shi & W. Wang, editors: Logic, Rationality, and Interaction, Springer Nature Singapore, pp. 110–123, doi:10.1007/978-981-95-2481-5_8

    work page doi:10.1007/978-981-95-2481-5_8 2025