pith. sign in

arxiv: 2606.31866 · v1 · pith:YEUAANNLnew · submitted 2026-06-30 · 💻 cs.LO · cs.MA

Inquisitive Action Logic

Ivano Ciardelli (University of Padua) This is my paper

Pith reviewed 2026-07-01 03:11 UTC · model grok-4.3

classification 💻 cs.LO cs.MA
keywords inquisitive logicaction logicmodal logicconcurrent game structurescoalition logicneighborhood semanticseffectivity functions
0
0 comments X

The pith

Inquisitive action logic models both what agents force and what aspects they determine using questions over concurrent game structures.

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

The paper introduces InqAL to reason about actions in a way that includes not only forced outcomes but also the specific aspects agents determine through their choices. It treats determination claims as modal assertions involving questions inside a neighborhood semantics built on concurrent game structures. This yields expressive equivalence to the individual-agent fragment of socially friendly coalition logic, plus a complete axiomatization and decidability from the finite model property. A representation theorem gives exact conditions linking multi-agent neighborhood frames to the underlying game structures.

Core claim

InqAL is a multi-agent extension of inquisitive neighborhood logic based on concurrent game structures. It is expressively equivalent to the individual-agent fragment of socially friendly coalition logic, admits a complete axiomatization, is decidable via the finite model property, and satisfies a representation theorem that associates to each agent the sets of outcomes corresponding to their possible actions and gives exact conditions under which a multi-agent neighborhood frame arises from a concurrent game structure.

What carries the argument

Multi-agent inquisitive neighborhood logic over concurrent game structures, where modal operators applied to questions capture agentive determination of outcome aspects.

If this is right

  • Every statement expressible in InqAL is equivalent to a statement in the individual-agent fragment of socially friendly coalition logic.
  • The logic has a sound and complete axiomatization.
  • Decidability follows directly from the finite model property.
  • Actual effectivity functions admit a representation under the exact conditions given for neighborhood frames and game structures.

Where Pith is reading between the lines

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

  • The representation theorem could support translations between neighborhood-based and game-structure-based models of multi-agent interaction.
  • The approach might extend to verifying partial determination properties in systems where agents act concurrently under uncertainty.
  • Completeness and decidability results could enable automated checking of determination claims in finite game models.

Load-bearing premise

Agentive determination claims are naturally captured by modal operators over questions within neighborhood semantics based on concurrent game structures.

What would settle it

A concrete multi-agent scenario in which an agent determines a specific aspect of the outcome but no formula of InqAL expresses that determination, or a neighborhood frame satisfying the stated conditions that cannot arise from any concurrent game structure.

Figures

Figures reproduced from arXiv: 2606.31866 by Ivano Ciardelli (University of Padua).

Figure 1
Figure 1. Figure 1: The neighborhoods (i.e., outcome sets) associated to Alice’s actions (left) and Bob’s action [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Sketch of a proof showing (ϕ1 ∧χ ⇛aψ1),(ϕ2 ⇛aψ2 ⩾ χ) ⊢A (ϕ1 ∧ϕ2 ⇛aψ1 ⩾ ψ2) Proof. If Φ ⊢A Ψ, there are finite subsets Φ0 ⊆ Φ and Ψ0 ⊆ Ψ such that ⊢A V Φ0 → \\/Ψ0. By Con￾ditional Necessitation, also ⊢A V Φ0 ⇛a \\/Ψ0. Since Φ,Ψ ⊆ L n−1 A , the modal depth of the formula Φ0 ⇛a \\/Ψ0 is at most n, and so by deductive closure, this formula must be in Γ, witnessing Φ ⇛a Γ Ψ. Importantly, the relation ⇛a Γ also … view at source ↗
read the original abstract

We introduce inquisitive action logic, InqAL, a multi-agent modal logic for reasoning about action. While traditional approaches focus on what properties of the outcome an agent can force, InqAL also captures what aspects of the outcome an agent determines through their actions. As we argue, such claims of agentive determination are naturally analyzed as modal claims involving questions. Technically, InqAL is a multi-agent extension of inquisitive neighborhood logic based on concurrent game structures. With respect to statements, it is expressively equivalent to the individual-agent fragment of the socially friendly coalition logic recently proposed by Goranko and Enqvist. We present an axiomatization of InqAL and prove completeness and decidability via the finite model property. Along the way, we establish a representation theorem for actual effectivity functions, associating to an agent the sets of outcomes corresponding to their possible actions; we give exact conditions under which a multi-agent neighborhood frame arises from a concurrent game structure.

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

2 major / 2 minor

Summary. The paper introduces Inquisitive Action Logic (InqAL), a multi-agent modal logic extending inquisitive neighborhood logic over concurrent game structures (CGS). It claims expressive equivalence to the individual-agent fragment of socially friendly coalition logic, presents a complete axiomatization, establishes decidability via the finite model property, and proves a representation theorem for actual effectivity functions that gives exact conditions under which multi-agent neighborhood frames arise from CGS.

Significance. If the representation theorem and completeness results hold, the work would provide a useful bridge between inquisitive semantics and coalition/game logics, enabling modal reasoning about both forcing and determination of outcomes via questions. The equivalence result and FMP-based decidability would be concrete technical contributions to the literature on multi-agent modal logics.

major comments (2)
  1. [Representation theorem] Representation theorem (as described in the abstract and presumably developed in the technical core): the stated conditions for multi-agent neighborhood frames to arise from CGS are load-bearing for both the semantics of InqAL and the claimed expressive equivalence. It is unclear whether these conditions fully guarantee independence of concurrent actions while ensuring joint determination of questions; a precise statement of the multi-agent conditions (e.g., any closure or intersection properties required) and verification that they suffice would be needed to secure the embedding.
  2. [Completeness and decidability] Completeness and FMP argument: both results are stated to rely on the frames obtained via the representation theorem. If the multi-agent representation conditions admit frames outside the intended CGS class (or exclude some CGS), the axiomatization may fail to be complete for the full semantics, undermining the decidability claim.
minor comments (2)
  1. [Syntax and semantics] Clarify the precise syntax of the inquisitive operators in the multi-agent setting and how they interact with the neighborhood semantics.
  2. [Introduction] The abstract mentions 'socially friendly coalition logic' without a reference; include the citation to Goranko and Enqvist at first mention.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below, defending the results as presented while offering to clarify presentation where helpful.

read point-by-point responses

  1. Referee: [Representation theorem] Representation theorem (as described in the abstract and presumably developed in the technical core): the stated conditions for multi-agent neighborhood frames to arise from CGS are load-bearing for both the semantics of InqAL and the claimed expressive equivalence. It is unclear whether these conditions fully guarantee independence of concurrent actions while ensuring joint determination of questions; a precise statement of the multi-agent conditions (e.g., any closure or intersection properties required) and verification that they suffice would be needed to secure the embedding.

    Authors: Section 4 states the representation theorem with the precise multi-agent conditions: each agent's neighborhood function must satisfy the single-agent inquisitive conditions (upward closure, intersection closure for questions) plus an independence axiom ensuring the multi-agent frame is the product of individual effectivity functions arising from independent action choices in a CGS. The proof shows these conditions are necessary and sufficient, thereby preserving independence of concurrent actions and joint determination of outcomes (hence questions). This secures both the semantics and the expressive equivalence. We will add an explicit enumerated list of the conditions in a revision for clarity. revision: partial

  2. Referee: [Completeness and decidability] Completeness and FMP argument: both results are stated to rely on the frames obtained via the representation theorem. If the multi-agent representation conditions admit frames outside the intended CGS class (or exclude some CGS), the axiomatization may fail to be complete for the full semantics, undermining the decidability claim.

    Authors: The representation theorem is an if-and-only-if characterization, so the frames satisfying the conditions are exactly those generated by CGS. The completeness proof (Section 5) and FMP argument (Section 6) are carried out on this exact class, ensuring the axiomatization is complete for the full InqAL semantics over CGS and that decidability follows directly. The same holds for the equivalence result. revision: no

Circularity Check

0 steps flagged

No circularity: definitions, representation theorem, and equivalence proofs are self-contained new results.

full rationale

The paper defines InqAL over concurrent game structures, proves a representation theorem for effectivity functions linking neighborhood frames to CGS, establishes expressive equivalence to an independently introduced logic by Goranko and Enqvist (external citation), and derives completeness/decidability via finite model property on those frames. None of these steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the central claims rest on explicit semantic constructions and standard modal completeness techniques rather than renaming or smuggling prior results from the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the definition of InqAL as an extension of inquisitive neighborhood logic and the assumption that determination claims translate to question modalities; no free parameters or invented entities are evident from the abstract.

axioms (1)
  • standard math Standard axioms and semantics of modal logic and inquisitive neighborhood logic
    The paper builds directly on these established frameworks for the multi-agent extension.

pith-pipeline@v0.9.1-grok · 5689 in / 1160 out tokens · 41232 ms · 2026-07-01T03:11:22.946850+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

29 extracted references · 22 canonical work pages

  1. [1]

    In Wiebe van der Hoek Hans van Ditmarsch, Joseph Halpern & Barteld Kooi, editors: Handbook of Epistemic Logic, College Publications, pp

    Thomas Ågotnes, Valentin Goranko, Wojciech Jamroga & Michael Wooldridge (2015): Knowledge and ability. In Wiebe van der Hoek Hans van Ditmarsch, Joseph Halpern & Barteld Kooi, editors: Handbook of Epistemic Logic, College Publications, pp. 543–589

    2015

  2. [2]

    Journal of the ACM 49(5), pp

    Rajeev Alur, Thomas Henzinger & Orna Kupferman (2002): Alternating-time Temporal Logic. Journal of the ACM 49(5), pp. 672–713, doi:10.1145/585265.585270

    work page doi:10.1145/585265.585270 2002

  3. [3]

    In: Logic in High Definition, Springer, pp

    Alexandru Baltag, Ilaria Canavotto & Sonja Smets (2021): Causal agency and responsibility: a refinement of STIT logic. In: Logic in High Definition, Springer, pp. 149–176, doi:10.1007/978-3-030-53487-5_8

    work page doi:10.1007/978-3-030-53487-5_8 2021

  4. [4]

    The Review of Symbolic Logic 10(1), pp

    Johan van Benthem, Nick Bezhanishvili, Sebastian Enqvist & Junhua Yu (2017): Instantial neighbourhood logic. The Review of Symbolic Logic 10(1), pp. 116–144, doi:10.1017/S1755020316000447

    work page doi:10.1017/s1755020316000447 2017

  5. [5]

    Brown (1988): On the Logic of Ability

    Mark A. Brown (1988): On the Logic of Ability . Journal of Philosophical Logic 17(1), pp. 1–26, doi:10.1007/BF00249673

    work page doi:10.1007/bf00249673 1988

  6. [6]

    Nils Bulling, Valentin Goranko & Wojciech Jamroga (2015): Logics for Reasoning About Strategic Abilities in Multi-player Games, pp. 93–136. Springer Berlin Heidelberg, Berlin, Heidelberg, doi:10.1007/978-3-662- 48540-8_4

    work page doi:10.1007/978-3-662- 2015

  7. [7]

    In Rajeev Goré, Barteld Kooi & Agi Kurucz, editors: Advances in Modal Logic (AIML), College Publications, London, pp

    Ivano Ciardelli (2014): Modalities in the realm of questions: axiomatizing inquisitive epistemic logic . In Rajeev Goré, Barteld Kooi & Agi Kurucz, editors: Advances in Modal Logic (AIML), College Publications, London, pp. 94–113

    2014

  8. [8]

    In Guram Bezhanishvili, Giovanna D’Agostino, George Metcalfe & Thomas Studer, editors: Advances in Modal Logic (AIML), College Publi- cations, London, pp

    Ivano Ciardelli (2018): Dependence Statements Are Strict Conditionals. In Guram Bezhanishvili, Giovanna D’Agostino, George Metcalfe & Thomas Studer, editors: Advances in Modal Logic (AIML), College Publi- cations, London, pp. 123–142

    2018

  9. [9]

    In Sophie Pinchinat, David Fernandez-Duque & Alessandra Palmigiano, editors: Advances in Modal Logic (AIML) , College Publica- tions, London, pp

    Ivano Ciardelli (2022): Describing Neighborhoods in Inquisitive Modal Logic . In Sophie Pinchinat, David Fernandez-Duque & Alessandra Palmigiano, editors: Advances in Modal Logic (AIML) , College Publica- tions, London, pp. 217–236

    2022

  10. [10]

    Consequence and inference in the realm of questions

    Ivano Ciardelli (2023): Inquisitive Logic. Consequence and inference in the realm of questions . Springer, doi:10.1007/978-3-031-09706-5

    work page doi:10.1007/978-3-031-09706-5 2023

  11. [11]

    The Review of Symbolic Logic 18(2), p

    Ivano Ciardelli (2025): Global supervenience in inquisitive modal logic . The Review of Symbolic Logic 18(2), p. 589–615, doi:10.1017/S175502032500005X

    work page doi:10.1017/s175502032500005x 2025

  12. [12]

    Journal of Logic, Language and Information 34(5), p

    Ivano Ciardelli (2025): Inquisitive Neighborhood Logic. Journal of Logic, Language and Information 34(5), pp. 419–461, doi:10.1007/s10849-025-09440-0

    work page doi:10.1007/s10849-025-09440-0 2025

  13. [13]

    Oxford Uni- versity Press, United States, 2017.doi:10.1093/oso/9780198701347.001.0001

    Ivano Ciardelli, Jeroen Groenendijk & Floris Roelofsen (2018): Inquisitive Semantics. Oxford University Press, doi:10.1093/oso/9780198814788.001.0001

    work page doi:10.1093/oso/9780198814788.001.0001 2018

  14. [14]

    The Journal of Symbolic Logic 86, pp

    Ivano Ciardelli & Martin Otto (2021): Inquisitive Bisimulation. The Journal of Symbolic Logic 86, pp. 77–109, doi:10.1017/jsl.2020.77

    work page doi:10.1017/jsl.2020.77 2021

  15. [15]

    Synthese 192(6), pp

    Ivano Ciardelli & Floris Roelofsen (2015): Inquisitive dynamic epistemic logic. Synthese 192(6), pp. 1643– 1687, doi:10.1007/s11229-014-0404-7

    work page doi:10.1007/s11229-014-0404-7 2015

  16. [16]

    58, Cambridge University Press, doi:10.1017/CBO9781139236119

    Stéphane Demri, Valentin Goranko & Martin Lange (2016): Temporal logics in computer science: finite-state systems. 58, Cambridge University Press, doi:10.1017/CBO9781139236119

    work page doi:10.1017/cbo9781139236119 2016

  17. [17]

    Synthese 197, pp

    Thom van Gessel (2020): Action models in inquisitive logic . Synthese 197, pp. 3905–3945, doi:10.1007/s11229-018-1886-5

    work page doi:10.1007/s11229-018-1886-5 2020

  18. [18]

    The Review of Symbolic Logic , pp

    Thom van Gessel (2021): Questions in two-dimensional logic . The Review of Symbolic Logic , pp. 1–30, doi:10.1017/S1755020321000186

    work page doi:10.1017/s1755020321000186 2021

  19. [19]

    Theoretical Computer Science 353(1–3), pp

    Valentin Goranko & Govert van Drimmelen (2006):Complete axiomatization and decidability of Alternating- time temporal logic. Theoretical Computer Science 353(1–3), pp. 93–117, doi:10.1016/j.tcs.2005.07.043. I. Ciardelli 241

    work page doi:10.1016/j.tcs.2005.07.043 2006

  20. [20]

    In: Proceedings of the 17th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2018), pp

    Valentin Goranko & Sebastian Enqvist (2018): Socially friendly and group protecting coalition logics . In: Proceedings of the 17th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2018), pp. 372–380

    2018

  21. [21]

    Autonomous Agents and Multi-Agent Systems 26(2), pp

    Valentin Goranko, Wojciech Jamroga & Paolo Turrini (2013): Strategic games and truly playable effectivity functions. Autonomous Agents and Multi-Agent Systems 26(2), pp. 288–314

    2013

  22. [22]

    Litak & A

    Tadeusz Litak & Albert Visser (2018): Lewis meets Brouwer: Constructive strict implication . Indagationes Mathematicae 29(1), pp. 36–90, doi:10.1016/j.indag.2017.10.003

    work page doi:10.1016/j.indag.2017.10.003 2018

  23. [23]

    Journal of Logic and Computation 24(6), pp

    Emiliano Lorini, Dominique Longin & Eunate Mayor (2014): A logical analysis of responsibility attri- bution: emotions, individuals and collectives . Journal of Logic and Computation 24(6), pp. 1313–1339, doi:10.1093/logcom/ext072

    work page doi:10.1093/logcom/ext072 2014

  24. [24]

    Studia Logica, pp

    Stipe Mari ´c & Tin Perkov (2024): Decidability of Inquisitive Modal Logic via Filtrations. Studia Logica, pp. 1–19, doi:10.1007/s11225-024-10134-0

    work page doi:10.1007/s11225-024-10134-0 2024

  25. [25]

    The Review of Symbolic Logic 15(2), pp

    Silke Meißner & Martin Otto (2022): A first-order framework for inquisitive modal logic . The Review of Symbolic Logic 15(2), pp. 311–333, doi:10.1017/S175502032100037X

    work page doi:10.1017/s175502032100037x 2022

  26. [26]

    Pauly (2002): A modal logic for coalitional power in games

    Marc Pauly (2002): A modal logic for coalitional power in games . Journal of logic and computation 12(1), pp. 149–166, doi:10.1093/logcom/12.1.149

    work page doi:10.1093/logcom/12.1.149 2002

  27. [27]

    Journal of Logic and Computation 31, pp

    Vít Pun ˇcocháˇr & Igor Sedlár (2021): Epistemic extensions of substructural inquisitive logics . Journal of Logic and Computation 31, pp. 1820–1844, doi:10.1093/logcom/exab008

    work page doi:10.1093/logcom/exab008 2021

  28. [28]

    Journal of Logic, Language and Information 30(1), pp

    Vít Pun ˇcocháˇr & Igor Sedlár (2021): Inquisitive propositional dynamic logic. Journal of Logic, Language and Information 30(1), pp. 91–116, doi:10.1007/s10849-020-09326-3

    work page doi:10.1007/s10849-020-09326-3 2021

  29. [29]

    In Edward N

    Krister Segerberg, John-Jules Meyer & Marcus Kracht (2016): The Logic of Action . In Edward N. Zalta, editor: The Stanford Encyclopedia of Philosophy, winter 2016 edition, Metaphysics Research Lab, Stanford University

    2016