Based on the intuitive idea that sets of objects or entities can be categorized in very different ways, and that some ways to categorise objects are better than others, depending on the purpose of the categorization, in this paper, a formal framework is introduced for parametrically generating a space of possible categorizations of a set of objects, based on the features which individual agents or groups thereof regard as relevant (formally encoded in the notion of interrogative agenda ). This formal framework accounts both for two-valued (crisp), and for many-valued (fuzzy) judgments about the relevance of given features, and introduces ways to aggregate individual agendas to group agendas. As an application on this framework, we discuss a machine-learning meta-algorithm for outlier detection and classification which provides local and global explanations of its results. • Parametric framework to generate categorization systems aligned with agent goals and knowledge stance. • Formal model of interrogative agendas with crisp and fuzzy importance judgments for feature relevance. • Operators to aggregate individual agendas into coherent group-level prioritization of categories. • FCA-based foundation for hierarchical, explainable, and uncertainty-aware categorization structures. • Meta-algorithm that learns agendas for classification, outlier detection, and explainable decision-making.

Categorization is one of the basic tasks in machine learning and data analysis. Building on formal concept analysis (FCA), the starting point of the present work is that different ways to categorize a given set of objects exist, which depend on the choice of the sets of features used to classify them, and different such sets of features may yield better or worse categorizations, relative to the task at hand. In their turn, the (a priori) choice of a particular set of features over another might be subjective and express a certain epistemic stance (e.g. interests, relevance, preferences) of an agent or a group of agents, namely, their interrogative agenda. In the present paper, we represent interrogative agendas as sets of features, and explore and compare different ways to categorize objects w.r.t. different sets of features (agendas). We first develop a simple unsupervised FCA-based algorithm for outlier detection which uses categorizations arising from different agendas. We then present a supervised meta-learning algorithm to learn suitable (fuzzy) agendas for categorization as sets of features with different weights or masses. We combine this meta-learning algorithm with the unsupervised outlier detection algorithm to obtain a supervised outlier detection algorithm. We show that these algorithms perform at par with commonly used algorithms for outlier detection on commonly used datasets in outlier detection. These algorithms provide both local and global explanations of their results.

The present paper establishes systematic connections among the first-order correspondents of Sahlqvist modal reduction principles in various relational semantic settings, including crisp and many-valued Kripke frames, and crisp and many-valued polarity-based frames (aka enriched formal contexts). Building on unified correspondence theory, we aim at introducing a theoretical environment which makes it possible to: (a) compare and inter-relate the various frame correspondents (in different relational settings) of any given Sahlqvist modal reduction principle; (b) recognize when first-order sentences in the frame-correspondence languages of different types of relational structures encode the same “modal content”; (c) meaningfully transfer and represent well known relational properties such as reflexivity, transitivity, symmetry, seriality, confluence, density, across different semantic contexts. These results can be understood as a first step in a research program aimed at making correspondence theory not just (methodologically) unified, but also (effectively) parametric.

The best known modal logics are axiomatized by Sahlqvist axioms, i.e., axioms of a syntactic shape which guarantees these formulas to have such excellent properties as canonicity and elementarity. Recently, the definition of Sahlqvist formulas has been generalized and extended from formulas in classical modal logic to inequalities (sequents) in a wide family of logics known as LE-logics. We introduce an algorithm which checks if a given inequality is generalized Sahlqvist in polynomial time.

In this article, we extend the research programme in algebraic proof theory from axiomatic extensions of the full Lambek calculus to logics algebraically captured by certain varieties of normal lattice expansions (normal LE-logics). Specifically, we generalize the residuated frames in Reference [ 34 ] to arbitrary signatures of normal lattice expansions (LE). Such a generalization provides a valuable tool for proving important properties of LE-logics in full uniformity. We prove semantic cut elimination for the display calculi \(\mathrm{D.LE}\) associated with the basic normal LE-logics and their axiomatic extensions with analytic inductive axioms. We also prove the finite model property (FMP) for each such calculus \(\mathrm{D.LE}\) , as well as for its extensions with analytic structural rules satisfying certain additional properties.

Abstract We define LE- $$\mathcal {ALC}$$ , a generalization of the description logic $$\mathcal {ALC}$$ based on the propositional logic of general (i.e. not necessarily distributive) lattices, and semantically interpreted on relational structures based on formal contexts from Formal Concept Analysis (FCA). The description logic LE- $$\mathcal {ALC}$$ allows us to formally describe databases with objects, features, and formal concepts, represented according to FCA as Galois-stable sets of objects and features. We describe ABoxes and TBoxes in LE- $$\mathcal {ALC}$$ , provide a tableaux algorithm for checking the consistency of LE- $$\mathcal {ALC}$$ knowledge bases with acyclic TBoxes, and show its termination, soundness and completeness. Interestingly, consistency checking for LE- $$\mathcal {ALC}$$ with acyclic TBoxes is in PTIME, while the complexity of the consistency checking of classical $$\mathcal {ALC}$$ with acyclic TBoxes is PSPACE-complete.

We introduce proper display calculi for intuitionistic, bi-intuitionistic and classical linear logics with exponentials, which are sound, complete, conservative, and enjoy cut elimination and subformula property. Based on the same design, we introduce a variant of Lambek calculus with exponentials, aimed at capturing the controlled application of exchange and associativity. Properness (i.e., closure under uniform substitution of all parametric parts in rules) is the main technical novelty of the present proposal, allowing both for the smoothest proof of cut elimination and for the development of an overarching and modular treatment for a vast class of axiomatic extensions and expansions of intuitionistic, bi-intuitionistic, and classical linear logics with exponentials. Our proposal builds on an algebraic and order-theoretic analysis of linear logic and applies the guidelines of the multi-type methodology in the design of display calculi.

A recent strand of research in structural proof theory aims at exploring the notion of analytic calculi (i.e., those calculi that support general and modular proof-strategies for cut elimination) and at identifying classes of logics that can be captured in terms of these calculi. In this context, Wansing introduced the notion of proper display calculi as one possible design framework for proof calculi in which the analyticity desiderata are realized in a particularly transparent way. Recently, the theory of properly displayable logics (i.e., those logics that can be equivalently presented with some proper display calculus) has been developed in connection with generalized Sahlqvist theory (a.k.a. unified correspondence). Specifically, properly displayable logics have been syntactically characterized as those axiomatized by analytic inductive axioms , which can be equivalently and algorithmically transformed into analytic structural rules so the resulting proper display calculi enjoy a set of basic properties: soundness, completeness, conservativity, cut elimination, and the subformula property. In this context, the proof that the given calculus is complete w.r.t. the original logic is usually carried out syntactically , i.e., by showing that a (cut-free) derivation exists of each given axiom of the logic in the basic system to which the analytic structural rules algorithmically generated from the given axiom have been added. However, so far, this proof strategy for syntactic completeness has been implemented on a case-by-case base and not in general. In this article, we address this gap by proving syntactic completeness for properly displayable logics in any normal (distributive) lattice expansion signature. Specifically, we show that for every analytic inductive axiom a cut-free derivation can be effectively generated that has a specific shape, referred to as pre-normal form .

We prove an algebraic canonicity theorem for normal LE-logics of arbitrary signature, in a generalized setting in which the non-lattice connectives are interpreted as operations mapping tuples of elements of the given lattice to closed or open elements of its canonical extension. Interestingly, the syntactic shape of LE-inequalities which guarantees their canonicity in this generalized setting turns out to coincide with the syntactic shape of analytic inductive inequalities , which guarantees LE-inequalities to be equivalently captured by analytic structural rules of a proper display calculus. We show that this canonicity result connects and strengthens a number of recent canonicity results in two different areas: subordination algebras, and transfer results via Gödel-McKinsey-Tarski translations.

In this paper, we generalize the basic notions and results of Dempster-Shafer theory from predicates to formal concepts. Results include the representation of conceptual belief functions as inner measures of suitable probability functions, and a Dempster-Shafer rule of combination on belief functions on formal concepts.

In recent years, unified correspondence has been developed as a generalized Sahlqvist theory which applies uniformly to all signatures of normal and regular (distributive) lattice expansions. This includes a general definition of the Sahlqvist and inductive formulas and inequalities in every such signature, based on order theory. This definition covers in particular all (bi-)intuitionistic modal logics. The theory of these logics has been intensively studied over the past seventy years in connection with classical polyadic modal logics, using suitable versions of Goedel-McKinsey-Tarski translations as main tools. It is therefore natural to ask (1) whether a general perspective on Goedel-McKinsey-Tarski translations can be attained, also based on order-theoretic principles like those underlying the general definition of Sahlqvist and inductive formulas and inequalities, which accounts for the known Goedel-McKinsey-Tarski translations and applies uniformly to all signatures of normal (distributive) lattice expansions; (2) whether this general perspective can be used to transfer correspondence and canonicity theorems for Sahlqvist and inductive formulas and inequalities in all signatures described above under Goedel-McKinsey-Tarski translations. In the present paper, we set out to answer these questions. We answer (1) in the affirmative; as to (2), we prove the transfer of the correspondence theorem for inductive inequalities of arbitrary signatures of normal distributive lattice expansions. We also prove the transfer of canonicity for inductive inequalities, but only restricted to arbitrary normal modal expansions of bi-intuitionistic logic. We also analyze the difficulties involved in obtaining the transfer of canonicity outside this setting, and indicate a route to extend the transfer of canonicity to all signatures of normal distributive lattice expansions.

The present article contributes to the development of the mathematical theory of epistemic updates using the tools of duality theory. Here, we focus on Probabilistic Dynamic Epistemic Logic (PDEL). We dually characterize the product update construction of PDEL-models as a certain construction transforming the complex algebras associated with the given model into the complex algebra associated with the updated model. Thanks to this construction, an interpretation of the language of PDEL can be defined on algebraic models based on Heyting algebras. This justifies our proposal for the axiomatization of the intuitionistic counterpart of PDEL.

The present article aims at establishing formal connections between correspondence phenomena, well known from the area of modal logic, and the theory of display calculi, originated by Belnap.These connections have been seminally observed and exploited by Marcus Kracht, in the context of his characterization of the modal axioms (which he calls primitive formulas) which can be effectively transformed into 'analytic'structural rules of display calculi.In this context, a rule is 'analytic'if adding it to a display calculus preserves Belnap's cut-elimination theorem.In recent years, the state-of-the-art in correspondence theory has been uniformly extended from classical modal logic to diverse families of non-classical logics, ranging from (bi-)intuitionistic (modal) logics, linear, relevant and other substructural logics, to hybrid logics and mu-calculi.This generalization has given rise to a theory called unified correspondence, the most important technical tools of which are the algorithm ALBA, and the syntactic characterization of Sahlqvist-type classes of formulas and inequalities which is uniform in the setting of normal DLE-logics (logics the algebraic semantics of which is based on bounded distributive lattices).We apply unified correspondence theory, with its tools and insights, to extend Kracht's results and prove his claims in the setting of DLE-logics.The results of the present article characterize the space of properly displayable DLE-logics.

We establish a formal connection between algorithmic correspondence theory and certain dual characterization results for finite lattices, similar to Nation's characterization of a hierarchy of pseudovarieties of finite lattices, progressively generalizing finite distributive lattices. This formal connection is mediated through monotone modal logic. Indeed, we adapt the correspondence algorithm ALBA to the setting of monotone modal logic, and we use a certain duality-induced encoding of finite lattices as monotone neighbourhood frames to translate lattice terms into formulas in monotone modal logic.

We extend unified correspondence theory to Kripke frames with impossible worlds and their associated regular modal logics. These are logics the modal connectives of which are not required to be normal: only the weaker properties of additivity |$\Diamond x\vee \Diamond y = \Diamond (x\vee y)$| and multiplicativity |$\Box x\wedge \Box y = \Box (x\wedge y)$| are required. Conceptually, it has been argued that their lacking necessitation makes regular modal logics better suited than normal modal logics at the formalization of epistemic and deontic settings. From a technical viewpoint, regularity proves to be very natural and adequate for the treatment of algebraic canonicity Jónsson-style. Indeed, additivity and multiplicativity turn out to be key to extend Jónsson’s original proof of canonicity to the full Sahlqvist class of certain regular distributive modal logics naturally generalizing distributive modal logic. Most interestingly, additivity and multiplicativity are key to Jónsson-style canonicity also in the original (i.e. normal DML. Our contributions include: the definition of Sahlqvist inequalities for regular modal logics on a distributive lattice propositional base; the proof of their canonicity following Jónsson’s strategy; the adaptation of the algorithm ALBA to the setting of regular modal logics on two non-classical (distributive lattice and intuitionistic) bases; the proof that the adapted ALBA is guaranteed to succeed on a syntactically defined class which properly includes the Sahlqvist one; finally, the application of the previous results so as to obtain proofs, alternative to Kripke’s, of the strong completeness of Lemmon’s epistemic logics E2-E5 with respect to elementary classes of Kripke frames with impossible worlds.

The theory of canonical extensions typically considers extensions of maps A→B to maps Aδ→Bδ. In the present article, the theory of canonical extensions of maps A→Bδ to maps Aδ→Bδ is developed, and is applied to obtain a new canonicity proof for those inequalities in the language of Distributive Modal Logic (DML) on which the algorithm ALBA [9] is successful.

We introduce a multi-type display calculus for Propositional Dynamic Logic (PDL). This calculus is complete w.r.t. PDL, and enjoys Belnap-style cut-elimination and subformula property.

The present article provides an analysis of the existing proof systems for dynamic epistemic logic from the viewpoint of proof-theoretic semantics. Dynamic epistemic logic is one of the best known members of a family of logical systems that have been successfully applied to diverse scientific disciplines, but the proof-theoretic treatment of which presents many difficulties. After an illustration of the proof-theoretic semantic principles most relevant to the treatment of logical connectives, we turn to illustrating the main features of display calculi, a proof-theoretic paradigm that has been successfully employed to give a proof-theoretic semantic account of modal and substructural logics. Then, we review some of the most significant proposals of proof systems for dynamic epistemic logics, and we critically reflect on them in the light of the previously introduced proof-theoretic semantic principles. The contributions of the present article include a generalization of Belnap's cut-elimination metatheorem for display calculi, and a revised version of the display-style calculus D.EAK [30]. We verify that the revised version satisfies the previously mentioned proof-theoretic semantic principles, and show that it enjoys cut-elimination as a consequence of the generalized metatheorem.

In the present article, we introduce a multi-type display calculus for dynamic epistemic logic, which we refer to as Dynamic Calculus. The display approach is suitable to modularly chart the space of dynamic epistemic logics on weaker-than-classical propositional base. The presence of types endows the language of the Dynamic Calculus with additional expressivity, allows for a smooth proof-theoretic treatment, and paves the way towards a general methodology for the design of proof systems for the generality of dynamic logics, and certainly beyond dynamic epistemic logic. We prove that the Dynamic Calculus adequately captures Baltag–Moss–Solecki's dynamic epistemic logic, and enjoys Belnap-style cut elimination.

Several researchers have collaborated to publish an overview of coalgebra and logic in the special issue of the Journal of Logic and Computation. They have informed that the idea of coalgebra is general enough to encompass structures that are not usually perceived as relational structures or transition systems. A coalgebra ξ resembles a topological space for TX=(PX)(2x) and helps in obtaining Chellas's conditional frames. Two states are defined in such a coalgebra to be behaviorally equivalent when they can be identified by some coalgebra morphism. This means in the case of deterministic automata that the two states induce the same accepted language. It is also observed that satisfiability of coalgebraic logic can be established in PSPACE and that complete coalgebraic logics have the finite model property.

Abstract In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.