
63Dynamic topological logic of metric spacesJournal of Symbolic Logic 77 (1): 308328. 2012.Dynamic Topological Logic ( $\mathcal{DTL}$ ) is a modal framework for reasoning about dynamical systems, that is, pairs 〈X, f〉 where X is a topological space and f: X → X a continuous function. In this paper we consider the case where X is a metric space. We first show that any formula which can be satisfied on an arbitrary dynamic topological system can be satisfied on one based on a metric space; in fact, this space can be taken to be countable and have no isolated points. Since any metric sp…Read more

61Evidence and plausibility in neighborhood structuresAnnals of Pure and Applied Logic 165 (1): 106133. 2014.The intuitive notion of evidence has both semantic and syntactic features. In this paper, we develop an evidence logic for epistemic agents faced with possibly contradictory evidence from different sources. The logic is based on a neighborhood semantics, where a neighborhood N indicates that the agent has reason to believe that the true state of the world lies in N. Further notions of relative plausibility between worlds and beliefs based on the latter ordering are then defined in terms of this …Read more

51On Provability Logics with Linearly Ordered ModalitiesStudia Logica 102 (3): 541566. 2014.We introduce the logics GLP Λ, a generalization of Japaridze’s polymodal provability logic GLP ω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP ω yielding among other things a finitary proof of the normal form theorem for the variablefree fragment of GLP Λ and the decidability of GLP Λ for recursive orderings Λ. Further, we give a restricted axiomatization of the variablefree frag…Read more

50Hyperations, Veblen progressions and transfinite iteration of ordinal functionsAnnals of Pure and Applied Logic 164 (78): 785801. 2013.Ordinal functions may be iterated transfinitely in a natural way by taking pointwise limits at limit stages. However, this has disadvantages, especially when working in the class of normal functions, as pointwise limits do not preserve normality. To this end we present an alternative method to assign to each normal function f a family of normal functions Hyp[f]=〈fξ〉ξ∈OnHyp[f]=〈fξ〉ξ∈On, called its hyperation, in such a way that f0=idf0=id, f1=ff1=f and fα+β=fα∘fβfα+β=fα∘fβ for all α, β.Hyperation…Read more

49Complete Intuitionistic Temporal Logics for Topological DynamicsJournal of Symbolic Logic 87 (3): 9951022. 2022.The language of linear temporal logic can be interpreted on the class of dynamic topological systems, giving rise to the intuitionistic temporal logic ${\sf ITL}^{\sf c}_{\Diamond \forall }$, recently shown to be decidable by FernándezDuque. In this article we axiomatize this logic, some fragments, and prove completeness for several familiar spaces.

47Tangled modal logic for topological dynamicsAnnals of Pure and Applied Logic 163 (4): 467481. 2012.

46The omegarule interpretation of transfinite provability logicAnnals of Pure and Applied Logic 169 (4): 333371. 2018.

44Dynamic Topological Logic Interpreted over Minimal SystemsJournal of Philosophical Logic 40 (6): 767804. 2011.Dynamic Topological Logic ( ) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems , which are pairs consisting of a topological space X and a continuous function f : X → X . The function f is seen as a change in one unit of time; within one can model the longterm behavior of such systems as f is iterated. One class of dynamic topological systems where the longterm behavior of f is particularly interesting is that of minimal systems ; …Read more

41Use case cards: a use case reporting framework inspired by the European AI ActEthics and Information Technology 26 (2): 123. 2024.Despite recent efforts by the Artificial Intelligence (AI) community to move towards standardised procedures for documenting models, methods, systems or datasets, there is currently no methodology focused on use cases aligned with the riskbased approach of the European AI Act (AI Act). In this paper, we propose a new framework for the documentation of use cases that we call use case cards, based on the use case modelling included in the Unified Markup Language (UML) standard. Unlike other docum…Read more

40Ecidence Logic: A New Look at Neighborhood StructuresIn Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic, Csli Publications. pp. 97118. 1998.

37A sound and complete axiomatization for Dynamic Topological LogicJournal of Symbolic Logic 77 (3): 947969. 2012.Dynamic Topological Logic (DFH) is a multimodal system for reasoning about dynamical systems. It is defined semantically and, as such, most of the work done in the field has been modeltheoretic. In particular, the problem of finding a complete axiomatization for the full language of DFH over the class of all dynamical systems has proven to be quite elusive. Here we propose to enrich the language to include a polyadic topological modality, originally introduced by Dawar and Otto in a different c…Read more

36Evidence Logic: A New Look at Neighborhood StructuresIn Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic, Csli Publications. pp. 97118. 1998.

35A Walk with GoodsteinBulletin of Symbolic Logic 30 (1): 119. 2024.Goodstein’s principle is arguably the first purely numbertheoretic statement known to be independent of Peano arithmetic. It involves sequences of natural numbers which at first appear to diverge, but eventually decrease to zero. These sequences are defined relative to a notation system based on exponentiation for the natural numbers. In this article, we provide a selfcontained and modern analysis of Goodstein’s principle, obtaining some variations and improvements. We explore notions of optim…Read more

29The Dynamics of Epistemic Attitudes in ResourceBounded AgentsStudia Logica 107 (3): 457488. 2019.The paper presents a new logic for reasoning about the formation of beliefs through perception or through inference in nonomniscient resourcebounded agents. The logic distinguishes the concept of explicit belief from the concept of background knowledge. This distinction is reflected in its formal semantics and axiomatics: we use a nonstandard semantics putting together a neighborhood semantics for explicit beliefs and relational semantics for background knowledge, and we have specific axioms …Read more

28The polytopologies of transfinite provability logicArchive for Mathematical Logic 53 (34): 385431. 2014.Provability logics are modal or polymodal systems designed for modeling the behavior of Gödel’s provability predicate and its natural extensions. If Λ is any ordinal, the GödelLöb calculus GLPΛ contains one modality [λ] for each λ < Λ, representing provability predicates of increasing strength. GLPω has no nontrivial Kripke frames, but it is sound and complete for its topological semantics, as was shown by Icard for the variablefree fragment and more recently by Beklemishev and Gabelaia for t…Read more

26Predicativity through transfinite reflectionJournal of Symbolic Logic 82 (3): 787808. 2017.Let T be a secondorder arithmetical theory, Λ a wellorder, λ < Λ and X ⊆ ℕ. We use $[\lambda X]_T^{\rm{\Lambda }}\varphi$ as a formalization of “φ is provable from T and an oracle for the set X, using ωrules of nesting depth at most λ”.For a set of formulas Γ, define predicative oracle reflection for T over Γ ) to be the schema that asserts that, if X ⊆ ℕ, Λ is a wellorder and φ ∈ Γ, then$$\forall \,\lambda < {\rm{\Lambda }}\,.$$In particular, define predicative oracle consistency ) as Pred…Read more

23Wellorders in the transfinite Japaridze algebraLogic Journal of the IGPL 22 (6): 933963. 2014.

23Strong completeness of provability logic for ordinal spacesJournal of Symbolic Logic 82 (2): 608628. 2017.

23Models of transfinite provability logicJournal of Symbolic Logic 78 (2): 543561. 2013.For any ordinal $\Lambda$, we can define a polymodal logic $\mathsf{GLP}_\Lambda$, with a modality $[\xi]$ for each $\xi < \Lambda$. These represent provability predicates of increasing strength. Although $\mathsf{GLP}_\Lambda$ has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variablefree fragment with natural number modalities, denoted $\mathsf{GLP}^0_\omega$. Later, Icard defined a topological model for $\mathsf{GLP}^0_\omega$ which is very closely rel…Read more

22Tableaux for structural abductionLogic Journal of the IGPL 20 (2): 388399. 2012.In this work, we shall study structural abduction and how ways of searching for solutions to the corresponding abductive problems could be modeled. Specifically, we shall define modal semantic tableaux for normal modal systems and study its applications to structural abduction. This method even makes structural abduction clearer and, as it shall be seen, when a radical change of logic is epistemologically required, the corresponding tableau will have pertinent information to suggest it

20Finiteness classes arising from Ramseytheoretic statements in set theory without choiceAnnals of Pure and Applied Logic 172 (6): 102961. 2021.

20Nondeterministic semantics for dynamic topological logicAnnals of Pure and Applied Logic 157 (23): 110121. 2009.Dynamic Topological Logic () is a combination of , under its topological interpretation, and the temporal logic interpreted over the natural numbers. is used to reason about properties of dynamical systems based on topological spaces. Semantics are given by dynamic topological models, which are tuples , where is a topological space, f a function on X and V a truth valuation assigning subsets of X to propositional variables. Our main result is that the set of valid formulas of over spaces with co…Read more

20Framevalidity Games and Lower Bounds on the Complexity of Modal AxiomsLogic Journal of the IGPL 30 (1): 155185. 2022.We introduce frameequivalence games tailored for reasoning about the size, modal depth, number of occurrences of symbols and number of different propositional variables of modal formulae defining a given frame property. Using these games, we prove lower bounds on the above measures for a number of wellknown modal axioms; what is more, for some of the axioms, we show that they are optimal among the formulae defining the respective class of frames.

19Taming the ‘Elsewhere’: On Expressivity of Topological LanguagesReview of Symbolic Logic 17 (1): 144153. 2024.In topological modal logic, it is well known that the Cantor derivative is more expressive than the topological closure, and the ‘elsewhere’, or ‘difference’, operator is more expressive than the ‘somewhere’ operator. In 2014, Kudinov and Shehtman asked whether the combination of closure and elsewhere becomes strictly more expressive when adding the Cantor derivative. In this paper we give an affirmative answer: in fact, the Cantor derivative alone can define properties of topological spaces not…Read more

19Verification logic: An arithmetical interpretation for negative introspectionIn Lev Beklemishev, Stéphane Demri & András Máté (eds.), Advances in Modal Logic, Volume 11, Csli Publications. pp. 120. 2016.

16Nonfinite Axiomatizability of Dynamic Topological LogicIn Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic, Csli Publications. pp. 200216. 1998.

14Absolute Completeness of S4u for Its MeasureTheoretic SemanticsIn Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic, Csli Publications. pp. 100119. 1998.

14Axiomatizing the lexicographic products of modal logics with linear temporal logicsIn Lev Beklemishev, Stéphane Demri & András Máté (eds.), Advances in Modal Logic, Volume 11, Csli Publications. pp. 7896. 2016.

14Hindman’s theorem in the hierarchy of choice principlesJournal of Mathematical Logic 24 (1). 2023.In the context of [Formula: see text], we analyze a version of Hindman’s finite unions theorem on infinite sets, which normally requires the Axiom of Choice to be proved. We establish the implication relations between this statement and various classical weak choice principles, thus precisely locating the strength of the statement as a weak form of the [Formula: see text].