Pith. sign in

REVIEW 4 major objections 5 minor 39 references

Axiomatics of Restricted Choices by Linear Orders of Sets with Minimum as Fallback

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves that, assuming the Axiom of Choice, a function on a restricted choice structure is representable by a K-minimal linear order over sets exactly when it satisfies the stated axioms, so restricting the possible outputs never…

desk verdict A real representation theorem for restricted choice functions with fallback; the proof is sound and the soft spots are presentational. read the letter →

arxiv 2506.03315 v2 pith:FFN6GVBE submitted 2025-06-03 cs.AI cs.LO

Kai Sauerwald , Kenneth Skiba , Eduardo Fermé , Thomas Meyer This is my paper
classification cs.AIcs.LO
keywords restrictedchoicestructuresfunctionslinearordersoversetsfallbackvalueaxiomatizationrevealedpreferencetheorychangeabstractargumentation
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

Choice functions normally pick a subset from each available menu, but many applications only allow certain 'realizable' outputs. This paper asks when such a restricted choice function can be represented by a linear order on the sets of alternatives, with a fixed fallback value K sitting at the bottom of the order. The answer is a full axiomatization: a function is representable exactly when it satisfies the postulates (SS0)-(SS4), (SS5E), and (SS6E), and for union-closed menus exactly when it satisfies the simpler (SS0)-(SS6). The point is that restricting the codomain of a choice function does not by itself destroy order-based representability, as long as an unrealizable request can be answered with the fallback minimum.

What carries the argument

The central object is the K-minimal linear choice function, defined from an R-smooth linear order ≪ on the realizable sets E by ▽^K_≪(S) = E when the unique ≪-minimal element of the realizable subsets of S is E, and ▽^K_≪(S) = K when there is none; R-smoothness means every menu that contains any realizable subset has a unique minimal realizable subset under the order. The argument is carried by a revealed-preference relation on the image of ▽: E1 ≼ E2 holds when there is a menu containing E1 ∪ E2 from which E1 is chosen (in the union-closed case, when ▽(E1 ∪ E2) = E1). This relation is reflexive, antisymmetric, consistent, and R-smooth; the proof extends it to a total preorder via an order-extension theorem and then to a linear order, with the fallback K placed as the unique global minimum. The postulates (SS5) and (SS5E) are the acyclicity condition that prevents cycles in this revealed relation, and (SS6)/(SS6E) makes the chosen set prevail over its own subsets.

What would settle it

Take a small finite union-closed restricted choice structure (for instance, a three-element alternative set with S = P(A) and a finite E), enumerate every function ▽: S → E that satisfies (SS0)-(SS6), and compare it against every function induced by a K-minimal linear order on E. If any axiom-satisfying function were not induced by some such order, Theorem 5.1 would be false; a brute-force check over the finitely many linear orders would settle it.

Watch

Extended reading notes

Core claim

Restrictions on the output side of a choice function usually break representation by a preference relation on alternatives, but not representation by a linear order on sets. The central result is a representation theorem: under the Axiom of Choice, for a normal restricted choice structure R = ⟨A, S, E⟩ with K ∈ E, a function ▽: S → E is a K-minimal linear choice function if and only if it satisfies (SS0)-(SS4) together with the two variants (SS5E) and (SS6E); when S is closed under unions, the simpler (SS0)-(SS6) are equivalent. The construction first encodes ▽ as a revealed-preference relation among realizable sets, proves that relation is consistent and smooth, and then extends it by an order-extension theorem to a linear order whose unique global minimum is K, with the choice from a menu read as the minimal realizable subset of that menu and K as the answer when no realizable subset exists. The existence half (Proposition 4.3) assures that some K-minimal linear choice function exists for every restricted choice structure, so the framework is never empty.

Load-bearing premise

The load-bearing premise is that the Axiom of Choice holds, so any consistent revealed-preference relation can be extended to a linear order; if Choice fails for infinite sets, the sufficiency direction may fail, and the paper also only covers normal structures where every realizable output is itself a possible input.

Editorial extensions

If this is right

  • Every restricted choice structure admits at least one K-minimal linear choice function for every K in E, so the fallback-based order representation always exists (Proposition 4.3).
  • For union-closed domains, the postulates (SS0)-(SS6) give a complete check: a function can be realized by a set-order with fallback exactly when it passes those six conditions.
  • In theory change, choice-based revision operators that are linear are exactly those satisfying (LCR1)-(LCR6), so the theorems transfer to a postulate-level account of revision when some outputs are not realizable.
  • In abstract argumentation, choice-based extension semantics that are linear are exactly those satisfying (LCA1)-(LCA6), giving a new semantics family that generalizes extension selection.
  • The fallback being the global minimum forces idempotence-like behavior: whenever the input contains K, the output must be K itself (SS2).

Reading between the lines

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

  • Extending the representation theorem beyond normal structures (dropping E ⊆ S) would require deciding how to make the fallback itself well-defined as an output that is not a legal input; the current proof does not address that case, so the first test of robustness is whether a variant of (SS1)-(SS2) can handle it.
  • On finite domains the axiomatization suggests an algorithm: check the postulates, build the revealed relation, and extend it to a linear order; violations of (SS5)/(SS5E) would show up directly as cycles, making acyclicity the only thing between a postulate-satisfying function and an explicit representing order.
  • Because K is the least preferred element, applications that want a different default behavior—such as the empty extension in argumentation—would need to move the fallback out of the minimum position or modify the axioms; the paper itself flags this as a limitation for argumentation semantics.
  • The same set-ordering technique with a fallback may transfer to preferential nonmonotonic reasoning, since the smoothness condition used here is a close relative of the smoothness condition in preferential semantics.
Share X Bluesky LinkedIn Reddit HN

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 introduces restricted choice structures R = <A, S, E> with E ⊆ S ⊆ P(A), and choice functions C : S → E that may return a fixed fallback K when no realizable subset of the input exists. Its central contribution is a representation theory for such functions by linear orders on E whose global minimum is K: Theorem 5.1 axiomatizes K-minimal linear choice functions on union-closed domains by (SS0)–(SS6), and Theorem 5.2 gives the general axiomatization (SS0)–(SS4), (SS5E), (SS6E). Both representation theorems are proved in the supplement via a revealed-preference encoding, Suzumura's theorem, and a tie-break to a linear order, with the Axiom of Choice explicitly assumed for the sufficiency direction. The paper also sketches applications to theory change and abstract argumentation.

Significance. If the results hold, the paper provides a clean and fairly general answer to when a choice function with restricted codomain admits a linear-order representation with a fallback minimum, extending classical revealed-preference axiomatizations. The strengths are the complete proofs in the supplemental material, the explicit and honest dependence on AC, and the fact that the fallback K is an external parameter rather than a fitted quantity; the axioms are not data-dependent. The applications to theory change and abstract argumentation are suggestive rather than fully developed, but they illustrate the intended use of the framework. The main representation theorems appear sound; the issues I found are local typos and missing justifications that are fixable without changing the results.

major comments (4)
  1. [4, Proposition 4.3(b)] The displayed formula for the swapped linear order ≪ is not the intended K-minimal order. In the last two unions, {(K′,K) | K′≪′K} ∪ {(K,K′) | K≪′K′}, the two elements are kept in their original order instead of being exchanged, so after the deletion of all pairs involving K and K′ the minimum of ≪ is still the old minimum rather than K in general. The existence claim is true and can be proved by a one-line renaming argument, but the displayed definition must be corrected or replaced.
  2. [Supplemental Material, Lemma A.6] The proof asserts that for every E ∈ Image(▽) there is S ∈ S with ▽(S) = E and S ⊆ E, but the subsequent application of (SS6E) with S1 = S3 = E and S2 = S requires E ∪ S = S, i.e., E ⊆ S. As written, the proof of Image(▽) = {E ∈ E | ▽(E) = E} is invalid. The needed inclusion E ⊆ S follows from (SS1) when E ≠ K and from S = K when E = K, so the lemma is correct after this typo is fixed; however, since the lemma is used to prove reflexivity of the R-compatible relation in Proposition A.8, the correction belongs in the main proof.
  3. [6.2, Theorem 6.6, (LCA2)] Postulate (LCA2) is stated as 'If KF ⊆ ΠF(E), then ΠF(E) = KF'. This is not the counterpart of (SS2), which concerns the input set: 'If K ⊆ S, then ▽(S) = K'. The theorem's proof imports (SS2) from Theorem 5.1, so (LCA2) should read 'If KF ⊆ E, then ΠF(E) = KF'. As printed, the postulate is a different condition; the authors should either correct it or justify that the printed version is equivalent in the presence of the other postulates.
  4. [6.1, Theorem 6.2 (and correspondingly 6.2, Theorem 6.6)] The proof of the converse direction claims that (SS0) follows from (SS1) and from the fact that EK contains only elements in the image of the operator. That inference is not immediate and, as written, is insufficient. A correct argument must first establish idempotence on the image: for E = K>S, show K>E = E (using (LCR6) with S1 = E and S2 = S), and then, when E ⊆ S, apply (LCR4) to conclude K>S ⊆ S. The same missing step occurs in Theorem 6.6. Please expand these two proofs.
minor comments (5)
  1. [Supplemental Material, Theorem 5.2 proof] The left-to-right direction of Theorem 5.2 cites Corollary A.4, but Corollary A.4 is stated only for union-closed restricted choice structures; the citation should be to Proposition A.3.
  2. [Supplemental Material, Proposition A.9] The step 'from < ⊆ ≪ we obtain min(S,⪯) ⊆ min(S,≪)' needs an explicit justification, because a tie in the total preorder ⊑ could in principle be broken by the tie-break in the wrong direction. In this setting the equality min(S,⊑) = min(S,⪯) and the fact that the latter is a singleton rule this out, but this should be stated.
  3. [Supplemental Material, Definition A.5 and Proposition A.9] The symbol E is used both for the full set of realizable choices and for the subdomain of the constructed relation; please use distinct symbols to avoid confusion.
  4. [4, Proposition 4.3(b) proof] There is a typo in the proof: 'well-wounded' should be 'well-founded'.
  5. [Abstract] The abstract contains a missing-space typo: 'Theaxiomaticsofsuchchoicefunctionsarepresented' should be 'The axiomatics of such choice functions are presented'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the representation theorems are proven from the postulates using external order-extension results, with no fitted parameter or self-citation carrying the load.

full rationale

The paper's central claim is a mathematical representation theorem, not an empirical prediction. Theorem 5.2 ('Assume the Axiom of Choice... A function ▽ : S → E is a K-minimal linear choice function for R if and only if the axioms (SS0)–(SS4), (SS5E) and (SS6E) are satisfied') is proved by two independent directions: Proposition A.3 verifies the postulates for every K-minimal linear choice function, and Propositions A.8–A.9 construct an R-compatible relation from a function satisfying the postulates, then extend it to a K-minimal R-smooth linear order via Suzumura's theorem (Theorem A.1) and the Linear Ordering Principle (Proposition A.2). The extension theorem is external to this paper and is not a self-citation. Theorem 5.1 then follows from Theorem 5.2 through Proposition A.10, which shows the equivalence of (SS5)/(SS6) and (SS5E)/(SS6E) under union-closure. The fallback K and the realizable set E are externally given inputs, not parameters fitted to the target result. Self-citations (e.g., [30], [32], [34], [36]) appear only as contextual references for applications and prior work; none is used to justify the representation theorems. The applications in Sections 6.1 and 6.2 instantiate Theorem 5.1, so any terse step in deriving (SS0) there is a proof-exposition issue rather than a circular reduction. No equation is defined in terms of the theorem's conclusion, and no known result is merely renamed.

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

No free parameters are fitted: the paper is a pure mathematical derivation with the fallback K as an externally given input and the postulates as the axiomatic base. The axioms listed are the background set-theoretic tools and domain assumptions the representation theorem relies on. No new entities are postulated.

assumptions (6)
  • standard math Axiom of Choice (ZFC)
    Stated in Theorems 5.1, 5.2 and Proposition 4.3; used to guarantee existence of well-founded linear orders and to invoke Suzumura's theorem for order extensions.
  • standard math Linear Ordering Principle
    Proposition A.2 assumes it to refine any total preorder into a linear order preserving the strict part; implied by AC.
  • standard math Suzumura's theorem
    Theorem A.1: consistency of a relation is equivalent to extendability to a total preorder; central to Proposition A.9.
  • domain assumption Normality of restricted choice structures: E subset of S
    Definition 3.1 restricts to normal structures; Section 3 explains that non-normal structures break chainability and fallback well-definedness.
  • domain assumption Fallback K encodes as unique global minimum (K-minimality)
    Definition 4.1 defines K-minimality and Definition 4.2 uses it; this is the specific fallback behavior that the axiomatization captures.
  • domain assumption R-smoothness of the representing order
    Definition 4.1: the order must have a minimum among the realizable subsets of each S whenever one exists; this is a required property of the representing order, not an additional assumption about the structure.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Axiomatics of Restricted Choices by Linear Orders of Sets with Minimum as Fallback." pith.science (2026). https://pith.science/paper/FFN6GVBE

@misc{pith2026250603315,
  author       = {Pith},
  title        = {Pith review of: Axiomatics of Restricted Choices by Linear Orders of Sets with Minimum as Fallback},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FFN6GVBE}},
  note         = {Machine review of arXiv:2506.03315}
}
read the original abstract

We study how linear orders can be employed to realise choice functions for which the set of potential choices is restricted, i.e., the possible choice is not possible among the full powerset of all alternatives. In such restricted settings, constructing a choice function via a relation on the alternatives is not always possible. However, we show that one can always construct a choice function via a linear order on sets of alternatives, even when a fallback value is encoded as the minimal element in the linear order. The axiomatics of such choice functions are presented for the general case and the case of union-closed input restrictions. Restricted choice structures have applications in knowledge representation and reasoning, and here we discuss their applications for theory change and abstract argumentation.

Figures

Figures reproduced from arXiv: 2506.03315 by the authors.

Figure 1
Figure 1. Illustration of the abstract argumentation framework from Example 6.7. [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

39 extracted references · 30 canonical work pages

  1. [1]

    The Journal of Symbolic Logic 50(2), 510–530 (1985)

    Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: Partial meet contraction and revision functions. The Journal of Symbolic Logic 50(2), 510–530 (1985)

    work page 1985

  2. [2]

    AI Mag.38(3), 25–36 (2017)

    Atkinson, K., Baroni, P., Giacomin, M., Hunter, A., Prakken, H., Reed, C., Simari, G.R., Thimm, M., Villata, S.: Towards artificial argumentation. AI Mag.38(3), 25–36 (2017). https://doi.org/10.1609/AIMAG.V38I3.2704

    doi:10.1609/aimag.v38i3.2704 2017

  3. [3]

    Studia Logica 107(5), 917–947 (2019).https://doi.org/10.1007/s11225-018-9812-x

    Baltag, A., Gierasimczuk, N., Smets, S.: Truth-tracking by belief revision. Studia Logica 107(5), 917–947 (2019).https://doi.org/10.1007/s11225-018-9812-x

    doi:10.1007/s11225-018-9812-x 2019

  4. [4]

    Baroni, P., Gabbay, D., Giacomin, M., Van der Torre, L.: Handbook of formal argumentation (2018)

    work page 2018

  5. [5]

    Artificial Intelligence171(10-15), 675–700 (2007).https: //doi.org/10.1016/J.ARTINT.2007.04.004

    Baroni, P., Giacomin, M.: On principle-based evaluation of extension-based ar- gumentation semantics. Artificial Intelligence171(10-15), 675–700 (2007).https: //doi.org/10.1016/J.ARTINT.2007.04.004

    doi:10.1016/j.artint.2007.04.004 2007

  6. [6]

    In: Destercke, S., Martinez, M.V., Sanfilippo, G

    Belahcène, K., Gaigne, J., Lagrue, S.: On decisive revision operators and to- tally ordered information. In: Destercke, S., Martinez, M.V., Sanfilippo, G. (eds.) Proceedings of the 16th International Conference on Scalable Uncertainty Man- agement (SUM 2024). LNCS, vol. 15350, pp. 45–52. Springer (2024). https: //doi.org/10.1007/978-3-031-76235-2_4

    work page doi:10.1007/978-3-031-76235-2_4 2024

  7. [7]

    In: Thielscher, M., Toni, F., Wolter, F

    Bonzon, E., Delobelle, J., Konieczny, S., Maudet, N.: Combining extension-based semantics and ranking-based semantics for abstract argumentation. In: Thielscher, M., Toni, F., Wolter, F. (eds.) Proceedings of the 16th International Conference, Principles of Knowledge Representation and Reasoning (KR 2018). pp. 118–127. AAAI Press (2018)

    work page 2018

  8. [8]

    Harvard University Press (2010)

    Bossert, W., Suzumura, K.: Consistency, choice, and rationality. Harvard University Press (2010)

    work page 2010

Show all 39 references
  1. [9]

    Journal of Computer and System Sciences80(2), 427–449 (2014)

    Creignou, N., Papini, O., Pichler, R., Woltran, S.: Belief revision within fragments of propositional logic. Journal of Computer and System Sciences80(2), 427–449 (2014). https://doi.org/10.1016/J.JCSS.2013.08.002

    2014 doi

  2. [10]

    Artificial Intelligence 218, 1–22 (2015).https://doi.org/10.1016/j.artint.2014.08.006

    Delgrande, J.P., Peppas, P.: Belief revision in Horn theories. Artificial Intelligence 218, 1–22 (2015).https://doi.org/10.1016/j.artint.2014.08.006

    2015 doi

  3. [11]

    Artificial Intelligence 77(2), 321–358 (1995).https://doi.org/10.1016/0004-3702(94)00041-X

    Dung, P.M.: On the acceptability of arguments and its fundamental role in non- monotonic reasoning, logic programming and n-person games. Artificial Intelligence 77(2), 321–358 (1995).https://doi.org/10.1016/0004-3702(94)00041-X

    1995 doi

  4. [12]

    In: Governatori, G., Turhan, A

    Falakh, F.M., Rudolph, S., Sauerwald, K.: Semantic characterizations of AGM revision for tarskian logics. In: Governatori, G., Turhan, A. (eds.) Rules and Rea- soning - 6th International Joint Conference on Rules and Reasoning, RuleML+RR 2022, Berlin, Germany, September 26-28,...

    2022 doi

  5. [13]

    Springer Briefs in Intelligent Systems, Springer (2018).https://doi.org/10.1007/ 978-3-319-60535-7

    Fermé, E.L., Hansson, S.O.: Belief Change - Introduction and Overview. Springer Briefs in Intelligent Systems, Springer (2018).https://doi.org/10.1007/ 978-3-319-60535-7

    2018

  6. [14]

    Studia Logica63(3), 331–342 (1999)

    Fermé, E.L., Hansson, S.O.: Selective revision. Studia Logica63(3), 331–342 (1999). https://doi.org/10.1023/A:1005294718935

    1999 doi

  7. [15]

    Artifi- cial Intelligence65(2), 197–245 (1994).https://doi.org/10.1016/0004-3702(94) 90017-5

    Gärdenfors, P., Makinson, D.: Nonmonotonic inference based on expectations. Artifi- cial Intelligence65(2), 197–245 (1994).https://doi.org/10.1016/0004-3702(94) 90017-5

    1994 doi

  8. [16]

    MIT Press Cambridge (1988) 16 Sauerwald, Skiba, Fermé, and Meyer

    Gärdenfors, P.: Knowledge in flux : modeling the dynamics of epistemic states. MIT Press Cambridge (1988) 16 Sauerwald, Skiba, Fermé, and Meyer

    1988

  9. [17]

    Erkenntnis50(2), 413–427 (1999)

    Hansson, S.O.: A survey of non-prioritized belief revision. Erkenntnis50(2), 413–427 (1999). https://doi.org/10.1023/A:1005534223776

    1999 doi

  10. [18]

    The Journal of Symbolic Logic66(4), 1581–1596 (2001)

    Hansson, S.O., Fermé, E.L., Cantwell, J., Falappa, M.A.: Credibility limited revision. The Journal of Symbolic Logic66(4), 1581–1596 (2001)

    2001

  11. [19]

    https://doi.org/10.34726/hss.2020.42785

    Haret, A.: Choosing what to believe: belief change through the lens of rational choice (2020). https://doi.org/10.34726/hss.2020.42785

    2020

  12. [20]

    Springer Monographs in Mathematics, Springer, Berlin, Germany, 3 edn

    Jech, T.J.: Set Theory. Springer Monographs in Mathematics, Springer, Berlin, Germany, 3 edn. (2002).https://doi.org/10.1007/3-540-44761-X

    2002 doi

  13. [21]

    In: Allen, J.F., Fikes, R., Sandewall, E

    Katsuno, H., Mendelzon, A.O.: On the difference between updating a knowledge base and revising it. In: Allen, J.F., Fikes, R., Sandewall, E. (eds.) Proceedings of the 2nd International Conference on Principles of Knowledge Representation and Reasoning (KR 1991). pp. 387–394. M...

    1991

  14. [22]

    Artificial Intelligence52(3), 263–294 (1991).https://doi.org/10.1016/ 0004-3702(91)90069-V

    Katsuno, H., Mendelzon, A.O.: Propositional knowledge base revision and minimal change. Artificial Intelligence52(3), 263–294 (1991).https://doi.org/10.1016/ 0004-3702(91)90069-V

    1991

  15. [23]

    Springer Science & Business Media (2013)

    Kelly, J.S.: Social choice theory: An introduction. Springer Science & Business Media (2013)

    2013

  16. [24]

    In: Destercke, S., Denoeux, T

    Konieczny, S., Marquis, P., Vesic, S.: On supported inference and extension selection in abstract argumentation frameworks. In: Destercke, S., Denoeux, T. (eds.) Pro- ceedings of the 13th European Conference Symbolic and Quantitative Approaches to Reasoning with Uncertainty (E...

    2015 doi

  17. [25]

    In: Brewka, G., Lang, J

    Konieczny, S., Pino Pérez, R.: Improvement operators. In: Brewka, G., Lang, J. (eds.) Proceedings of the 11th International Conference on Principles of Knowledge Representation and Reasoning (KR 2008). pp. 177–187. AAAI Press (2008)

    2008

  18. [26]

    Artificial Intelligence44(1-2), 167–207 (1990).https://doi

    Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence44(1-2), 167–207 (1990).https://doi. org/10.1016/0004-3702(90)90101-5

    1990 doi

  19. [27]

    Lehmann, D., Magidor, M.: What does a conditional knowledge base entail? Arti- ficial Intelligence 55(1), 1–60 (1992).https://doi.org/10.1016/0004-3702(92) 90041-U

    1992 doi

  20. [28]

    Studia Logica 52(3), 339–379 (1993)

    Makinson, D.: Five faces of minimality. Studia Logica 52(3), 339–379 (1993). https://doi.org/10.1007/BF01057652

    1993 doi

  21. [29]

    Oxford University Press, New York (2001)

    Rott, H.: Change, Choice and Inference: A Study of Belief Revision and Nonmono- tonic Reasoning. Oxford University Press, New York (2001)

    2001

  22. [30]

    In: Sanfilippo, G., Destercke, S., Martinez, V

    Sauerwald, K.: Credibility-limited revision for epistemic spaces. In: Sanfilippo, G., Destercke, S., Martinez, V. (eds.) Proceedings of the 16th International Conference on Scalable Uncertainty Management (SUM 2024) (2024).https://doi.org/10. 24963/kr.2022/42

    2024

  23. [31]

    In: Kern-Isberner, G., Ognjanovic, Z

    Sauerwald, K., Beierle, C.: Decrement operators in belief change. In: Kern-Isberner, G., Ognjanovic, Z. (eds.) Proceedings of the 15th European Conference Sym- bolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2019). LNCS, vol. 11726, pp. 251–262. Spring...

    2019

  24. [32]

    In: Gierasimczuk, N., Heyninck, J

    Sauerwald, K., Kontinen, J.: A first peek into preferential logics with team semantics. In: Gierasimczuk, N., Heyninck, J. (eds.) Proceedings of the 22nd International Workshop on Nonmonotonic Reasoning (NMR 2024) co-located with 21st Inter- national Conference on Principles o...

    2024

  25. [33]

    In: Casini, G., Dundua, B., Kutsia, T

    Sauerwald, K., Skiba, K., Fermé, E., Meyer, T.: Axiomatics of restricted choices by linear orders of sets with minimum as fallback. In: Casini, G., Dundua, B., Kutsia, T. (eds.) Logics in Artificial Intelligence. pp. 58–74. Springer Nature Switzerland, Cham (2025)

    2025

  26. [34]

    In: Marquis, P., Ortiz, M., Pagnucco, M

    Sauerwald, K., Thimm, M.: The realizability of revision and contraction operators in epistemic spaces. In: Marquis, P., Ortiz, M., Pagnucco, M. (eds.) Proceedings of the 21st International Conference on Principles of Knowledge Representation and Reasoning (KR 2024). pp. 665–67...

    2024 doi

  27. [35]

    The Review of Economic Studies 38(3), 307–317 (1971).https://doi.org/10.2307/2296384

    Sen, A.K.: Choice functions and revealed preference. The Review of Economic Studies 38(3), 307–317 (1971).https://doi.org/10.2307/2296384

    1971 doi

  28. [36]

    In: Zhou, Z

    Skiba, K., Rienstra, T., Thimm, M., Heyninck, J., Kern-Isberner, G.: Ranking extensions in abstract argumentation. In: Zhou, Z. (ed.) Proceedings of the 30th International Joint Conference on Artificial Intelligence (IJCAI 2021). pp. 2047–2053. ijcai.org (2021). https://doi.or...

    2021 doi

  29. [37]

    Economica43(172), 381–390 (1976)

    Suzumura, K.: Remarks on the theory of collective choice. Economica43(172), 381–390 (1976). https://doi.org/10.2307/2553273

    1976 doi

  30. [38]

    Journal of Logics and their Applications4(8), 2735–2778 (2017)

    van der Torre, L., Vesic, S.: The principle-based approach to abstract argumentation semantics. Journal of Logics and their Applications4(8), 2735–2778 (2017)

    2017

  31. [39]

    ⇒”] Corollary A.4 yields the left-to-right direction. [“⇐

    Zermelo, E.: Beweis, daß jede menge wohlgeordnet werden kann. Mathematische Annalen 59(4), 514–516 (1904).https://doi.org/10.1007/BF01445300 Preprint Disclaimer This preprint has not undergone any post-submission improvements or corrections. The Version of Record of this contr...

    1904 doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.