Showing papers on "Modal operator published in 2019"

Negation on the Australian Plan

Abstract: We present and defend the Australian Plan semantics for negation. This is a comprehensive account, suitable for a variety of different logics. It is based on two ideas. The first is that negation is an exclusion-expressing device: we utter negations to express incompatibilities. The second is that, because incompatibility is modal, negation is a modal operator as well. It can, then, be modelled as a quantifier over points in frames, restricted by accessibility relations representing compatibilities and incompatibilities between such points. We defuse a number of objections to this Plan, raised by supporters of the American Plan for negation, in which negation is handled via a many-valued semantics. We show that the Australian Plan has substantial advantages over the American Plan.

The Semiotic Construction of the Sense of Agency. The Modal Articulation in Narrative Processes.

Abstract: The sense of agency is an ongoing process of semiotic construction of the action starting from the affective, cognitive, intersubjective and cultural matrix of experience. A person narratively constructs the sense of her agentive experience and in doing so does not refer exclusively to the "what", but also to the "how". There is always a specific "modus" to experience one's own action. We present the psychological notion of the Modal Articulation Process (MAP), namely the way through which a person orients and configures in a contextual frame the sense of her actions by means of modal operators of necessity, possibility, impossibility, contingency, but also knowledge, will, capability, constrain and opportunity. The notion of Modal Articulation Process is proposed as a semiotic, dynamic and recursive process that articulates narratively many aspects of the agency: the relational positionings and the way of experiencing them, the constraints and the resources present in the socio-symbolic context, the inherent temporality of every human phenomenon. Although the study of modal operators has an ancient and solid tradition of research in the fields of modal logics, analytical philosophy and narrative semiotic disciplines as well, yet in the field of the psychological sciences - except for a few authoritative isolated cases (Kurt Lewin, Rom Harre, Jaan Valsiner) - there is not a great deal of attention on the relevance of these symbolic devices and their function in constructing the sense of action in a narrative way. Indeed modal articulation processes are at stake both during daily common routines and during exceptional turning point experiences that request a reconfiguration of the sense of one's own agency (e.g. the experiences of illness demand a new modal re-articulation). Our discussion is aimed at deepening and developing the notion of modal articulation, its functions and its specificities.

Logics for rough concept analysis

Abstract: Taking an algebraic perspective on the basic structures of Rough Concept Analysis as the starting point, in this paper we introduce some varieties of lattices expanded with normal modal operators which can be regarded as the natural rough algebra counterparts of certain subclasses of rough formal contexts, and introduce proper display calculi for the logics associated with these varieties which are sound, complete, conservative and with uniform cut elimination and subformula property. These calculi modularly extend the multi-type calculi for rough algebras to a ‘nondistributive’ (i.e. general lattice-based) setting.

Formula size games for modal logic and μ-calculus

Abstract: We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke-models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler-Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler-Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic $\mathrm{FO}$ and (basic) modal logic $\mathrm{ML}$. We also present a version of the game for the modal $\mu$-calculus $\mathrm{L}_\mu$ and show that $\mathrm{FO}$ is also non-elementarily more succinct than $\mathrm{L}_\mu$.

The modal logic of Bayesian belief revision

Abstract: In Bayesian belief revision a Bayesian agent revises his prior belief by conditionalizing the prior on some evidence using Bayes’ rule. We define a hierarchy of modal logics that capture the logical features of Bayesian belief revision. Elements in the hierarchy are distinguished by the cardinality of the set of elementary propositions on which the agent’s prior is defined. Inclusions among the modal logics in the hierarchy are determined. By linking the modal logics in the hierarchy to the strongest modal companion of Medvedev’s logic of finite problems it is shown that the modal logic of belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable.

A Henkin-style completeness proof for the modal logic S5

Abstract: This paper presents a recent formalization of a Henkin-style completeness proof for the propositional modal logic S5 using the Lean theorem prover. The proof formalized is close to that of Hughes and Cresswell [9], except that it is given for a system based on a different choice of axioms. Here the proof is based on a Hilbert-style presentation better described as a Mendelson system augmented with axiom schemes for K, T, S4, and B, and the necessitation rule as rule of inference. The language has the false and implication as the only primitive logical connectives and necessity as the only primitive modal operator. The full source code is available online and has been typechecked with Lean 3.4.1.

Duality Results for (Co)Residuated Lattices

Abstract: We present dualities (discrete duality, duality via truth and Stone duality) for implicative and (co)residuated lattices. In combination with our recent article on a discrete duality for lattices with unary modal operators, the present article contributes in filling in a gap in the development of Orlowska and Rewitzky’s research program of discrete dualities, which seemed to have stumbled on the case of non-distributive lattices with operators. We discuss dualities via truth, which are essential in relating the non-distributive logic of two-sorted frames with their sorted, residuated modal logic, as well as full Stone duality for (co)residuated lattices. Our results have immediate applications to the semantics of related substructural (resource consious) logical calculi.

Modal Boolean Connexive Logics: Semantics and Tableau Approach

Abstract: In this paper we investigate Boolean connexive logics in a language with modal operators: □, ◊. In such logics, negation, conjunction, and disjunction behave in a classical, Boolean way. Only implication is non-classical. We construct these logics by mixing relating semantics with possible worlds. This way, we obtain connexive counterparts of basic normal modal logics. However, most of their traditional axioms formulated in terms of modalities and implication do not hold anymore without additional constraints, since our implication is weaker than the material one. In the final section, we present a tableau approach to the discussed modal logics.

Perspectival control and obviation in directive clauses

Abstract: The paper proposes a new type of control configuration: perspectival control. This involves control of a non-argument PRO that combines with a directive modal operator in the Mood domain. This PRO encodes the individual to whom the public commitments associated with the modal are anchored, and its presence can be detected in the syntax through a subject obviation effect. The empirical focus of the paper are Slovenian directive clauses (imperatives and subjunctives), but the analysis is shown to also have implications for analyses of other languages, as well as theories of directive clauses and the representation of discourse-related information in the syntax.

Strong Noncontingency: On the Modal Logics of an Operator Expressively Weaker Than Necessity

Abstract: Operators can be compared in at least two respects: expressive strength and deductive strength. Inspired by Hintikka’s treatment of question embedding verbs, the variations of noncontingency operator, and also the various combinations of modal operators and Boolean connectives, we propose a logic with (deductively) strong noncontingency operator as the only primitive modality. The novel operator is deductively but not expressively stronger than both noncontingency operator and essence operator, and expressively but not deductively weaker than the necessity operator. The frame-definability power of this new logic is in between standard modal logic and noncontingency logic. A notion of bisimulation is proposed to characterize this logic within standard modal logic and first-order logic. Axiomatizations over various frame classes are presented, among which the minimal logic is related to the treatment of an alternative semantics of the agreement operator proposed by Lloyd Humberstone.

Term-Sequence-Modal Logics.

Abstract: Term-modal logics, developed by Fitting et al., enable us to index a modal operator by a term of the first-order logic and even to quantify variables in the index of the modal operator. In this paper, we expand term-modal logics by allowing a modal operator to be indexed by a finite sequence of terms as well as a single term. The expanded logics are generalizations of both term-modal logics and quantified modal logics. We provide sound Hilbert-style axiomatizations (without Barcan-like axioms) for the logics and establish the strong completeness results for some of the logics. We also propose sequent calculi for the logics and show cut elimination theorems and Craig interpolation theorems for some of the calculi.

Intuitionistic Non-Normal Modal Logics: A general framework

Abstract: We define a family of intuitionistic non-normal modal logics; they can bee seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only one between Necessity and Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. For all logics we provide both a Hilbert axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then give a semantic characterisation of our logics in terms of neighbourhood models. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K (CK) and the propositional fragment of Wijesekera's Constructive Concurrent Dynamic Logic.

The modality and non-extensionality of the quantifiers

Abstract: We shall try to defend two non-standard views that run counter to two well-entrenched familiar views. The standard views are (1) the universal and existential quantifiers of first-order logic are not modal operators, and (2) the quantifiers are extensional. If that is correct then the counterclaims create genuine problems for some traditional philosophical doctrines.

Reasoning about opinion dynamics in social networks

Abstract: This article introduces a logic to reason about a well-known model of opinion dynamics in social networks initially developed by Morris DeGroot as well as Keith Lehrer and Carl Wagner. The proposed logic is an extension of Łukasiewicz' fuzzy logic with additional equational expressivity, modal operators, machinery from hybrid logic and dynamic modalities. The model of opinion dynamics in social networks is simple enough to be easily grasped, but still complex enough to have interesting mathematical properties and applications. Thus, developing a logic to reason about this particular model serves as a paradigmatic example of how logic can be useful in social network analysis.

A modal logic for subject-oriented spatial reasoning

Abstract: We present a modal logic for representing and reasoning about space seen from the subject's perspective. The language of our logic comprises modal operators for the relations "in front", "behind", "to the left", and "to the right" of the subject, which introduce the intrinsic frame of reference; and operators for "behind an object", "between the subject and an object", "to the left of an object", and "to the right of an object", employing the relative frame of reference. The language allows us to express nominals, hybrid operators, and a restricted form of distance operators which, as we demonstrate by example, makes the logic interesting for potential applications. We prove that the satisfiability problem in the logic is decidable and in particular PSpace-complete.

Axiomatizing logics of fuzzy preferences using graded modalities

Abstract: The aim of this paper is to propose a many-valued modal framework to formalize reasoning with both graded preferences and propositions, in the style of van Benthem et al.'s classical modal logics for preferences. To do so, we start from Bou et al.'s minimal modal logic over a finite and linearly ordered residuated lattice. We then define appropriate extensions on a multi-modal language with graded modalities, both for weak and strict preferences, and with truth-constants. Actually, the presence of truth-constants in the language allows us to show that the modal operators Box and Diamond of the minimal modal logic are inter-definable. Finally, we propose an axiomatic system for this logic in an extended language (where the preference modal operators are definable), and prove completeness with respect to the intended graded preference semantics.

Birkhoff Completeness for Hybrid-Dynamic First-Order Logic

Abstract: Hybrid-dynamic first-order logic is a kind of modal logic obtained by enriching many-sorted first-order logic with features that are common to hybrid and to dynamic logics. This provides us with a logical system with an increased expressive power thanks to a number of distinctive attributes: first, the possible worlds of Kripke structures, as well as the nominals used to identify them, are endowed with an algebraic structure; second, we distinguish between rigid symbols, which have the same interpretation across possible worlds – and thus provide support for the standard rigid quantification in modal logic – and flexible symbols, whose interpretation may vary; third, we use modal operators over dynamic-logic actions, which are defined as regular expressions over binary nominal relations. In this context, we propose a general notion of hybrid-dynamic Horn clause and develop a proof calculus for the Horn-clause fragment of hybrid-dynamic first-order logic. We investigate soundness and compactness properties for the syntactic entailment system that corresponds to this proof calculus, and prove a Birkhoff-completeness result for hybrid-dynamic first-order logic.

Towards Contingent World Descriptions in Description Logics

Abstract: The philosophical, logical, and terminological junctions between Description Logics (DLs) and Modal Logic (ML) are important because they can support the formal analysis of modal notions of ‘possibility’ and ‘necessity’ through the lens of DLs. This paper introduces functional contingents in order to (i) structurally and terminologically analyse ‘functional possibility’ and ‘functional necessity’ in DL world descriptions and (ii) logically and terminologically annotate DL world descriptions based on functional contingents. The most significant contributions of this research are the logical characterisation and terminological analysis of functional contingents in DL world descriptions. The ultimate goal is to investigate how modal operators can – logically and terminologically – be expressed within DL world descriptions.

The Logic of Imagination Acts: A Formal System for the Dynamics of Imaginary Worlds

Abstract: Imagination has received a great deal of attention in different fields such as psychology, philosophy and the cognitive sciences, in which some works provide a detailed account of the mechanisms involved in the creation and elaboration of imaginary worlds. Although imagination has also been formalized using different logical systems, none of them captures those dynamic mechanisms. In this work, we take inspiration from the Common Frame for Imagination Acts, that identifies the different processes involved in the creation of imaginary worlds, and we use it to define a dynamic formal system called the Logic of Imagination Acts. We build our logic by using a possible-worlds semantics, together with a new set of static and dynamic modal operators. The role of the new dynamic operators is to call different algorithms that encode how the formal model is expanded in order to capture the different mechanisms involved in the creation and development of imaginary worlds. We provide the definitions of the language, the semantics and the algorithms, together with an example that shows how the model is expanded. By the end, we discuss some interesting features of our system, and we point out to possible lines of future work.

Canonical Models and the Complexity of Modal Team Logic

Abstract: We study modal team logic MTL, the team-semantical extension of modal logic ML closed under Boolean negation. Its fragments, such as modal dependence, independence, and inclusion logic, are well-understood. However, due to the unrestricted Boolean negation, the satisfiability problem of full MTL has been notoriously resistant to a complexity theoretical classification. In our approach, we introduce the notion of canonical models into the team-semantical setting. By construction of such a model, we reduce the satisfiability problem of MTL to simple model checking. Afterwards, we show that this approach is optimal in the sense that MTL-formulas can efficiently enforce canonicity. Furthermore, to capture these results in terms of complexity, we introduce a non-elementary complexity class, TOWER(poly), and prove that it contains satisfiability and validity of MTL as complete problems. We also prove that the fragments of MTL with bounded modal depth are complete for the levels of the elementary hierarchy (with polynomially many alternations). The respective hardness results hold for both strict or lax semantics of the modal operators and the splitting disjunction, and also over the class of reflexive and transitive frames.

Logicality, Double-Line Rules, and Modalities

Abstract: This paper deals with the question of the logicality of modal logics from a proof-theoretic perspective. It is argued that if Dosen’s analysis of logical constants as punctuation marks is embraced, it is possible to show that all the modalities in the cube of normal modal logics are indeed logical constants. It will be proved that the display calculus for each displayable modality admits a purely structural presentation based on double-line rules which, following Dosen’s analysis, allows us to claim that the corresponding modal operators are logical constants.

On the most extended interval-valued intuitionistic fuzzy modal operators from both types

Abstract: The most extended (by the moment) interval-valued intuitionistic fuzzy modal operators from both types are introduced. A theorem for equivalence of two of them is proved.

Modal operators on rings of continuous functions

Abstract: It is a classic result in modal logic that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality. Our goal is to further generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with the boolean algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main novelty is to define a modal operator on such a ring utilizing a continuous relation on a compact Hausdorff space. Our starting point is the well-known Gelfand duality between the category $KHaus$ of compact Hausdorff spaces and the category $ubal$ of uniformly complete bounded archimedean $\ell$-algebras. We endow a bounded archimedean $\ell$-algebra with a modal operator, which results in the category $mbal$ of modal bounded archimedean $\ell$-algebras. Our main result establishes a dual adjunction between $mbal$ and the category $KHK$ of what we call compact Hausdorff frames; that is, Kripke frames equipped with a compact Hausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between $KHK$ and the reflective subcategory $mubal$ of $mbal$ consisting of uniformly complete objects of $mbal$. This generalizes both Gelfand duality and the duality for modal algebras.

Monotonic Distributive Semilattices

Abstract: In the study of algebras related to non-classical logics, (distributive) semilattices are always present in the background. For example, the algebraic semantic of the {→, ∧, ⊤}-fragment of intuitionistic logic is the variety of implicative meet-semilattices (Chellas 1980; Hansen 2003). In this paper we introduce and study the class of distributive meet-semilattices endowed with a monotonic modal operator m. We study the representation theory of these algebras using the theory of canonical extensions and we give a topological duality for them. Also, we show how our new duality extends to some particular subclasses.

Uniform labelled calculi for conditional and counterfactual logics

Abstract: Lewis’s counterfactual logics are a class of conditional logics that are defined as extensions of classical propositional logic with a two-place modal operator expressing conditionality. Labelled proof systems are proposed here that capture in a modular way Burgess’s preferential conditional logic \( \mathbb {PCL}\), Lewis’s counterfactual logic \( \mathbb {V}\), and their extensions. The calculi are based on preferential models, a uniform semantics for conditional logics introduced by Lewis. The calculi are analytic, and their completeness is proved by means of countermodel construction. Due to termination in root-first proof search, the calculi also provide a decision procedure for the logics.

A Four-Valued Hybrid Logic with Non-dual Modal Operators

Abstract: Hybrid logics are an extension of modal logics where it is possible to refer to a specific state, thus allowing the description of what happens at specific states, equalities and transitions between them. This makes hybrid logics very desirable to work with relational structures.

Horn Clauses in Hybrid-Dynamic First-Order Logic

Abstract: We propose a hybrid-dynamic first-order logic as a formal foundation for specifying and reasoning about reconfigurable systems. As the name suggests, the formalism we develop extends (many-sorted) first-order logic with features that are common to hybrid logics and to dynamic logics. This provides certain key advantages for dealing with reconfigurable systems, such as: (a) a signature of nominals, including operation and relation symbols, that allows references to specific possible worlds / system configurations -- as in the case of hybrid logics; (b) distinguished signatures of rigid and flexible symbols, where the rigid symbols are interpreted uniformly across possible worlds; this supports a rigid form of quantification, which ensures that variables have the same interpretation regardless of the possible world where they are evaluated; (c) hybrid terms, which increase the expressive power of the logic in the context of rigid symbols; and (d) modal operators over dynamic-logic actions, which are defined as regular expressions over binary nominal relations. We then study Horn clauses in this hybrid-dynamic logic, and develop a series of results that lead to an initial-semantics theorem for arbitrary sets of clauses. This shows that a significant fragment of hybrid-dynamic first-order logic has good computational properties, and can serve as a basis for defining executable languages for reconfigurable systems. Lastly, we set out the foundations of logic programming in this fragment by proving a hybrid-dynamic variant of Herbrand's theorem, which reduces the semantic entailment of a logic-programming query by a program to the search of a suitable answer substitution.