Modal Operators Research Articles

Compositional Strategies for Large-Scale Works Around 1600

Giovanni Gabrieli’s posthumously published motets In ecclesiis and Benedictus es, Dominus ( Sacrae symphoniae ... liber secundus , 1615) show obvious characteristics of self-quotation and reworking. An analysis of the works focuses on a characteristic ascending-fifths sequence which has been called “Monte Romanesca” by Robert Gjerdingen who introduced the term for the eighteenth century. In this study, the term is expanded to include music from the end of the sixteenth century for the first time. Further analysis concentrates on motivic and modal connections as well as special harmonic features such as chromatic-third relationships. Additionally, several other works of the composer are considered, showing that Gabrieli built on highly sophisticated harmonic, motivic, and formal compositional strategies in his large-scale oeuvre. Also, apparent processes of imitation in the Missa sine nomine of Gabrieli’s pupil Alessandro Tadei suggest that the younger composer used either or both of his teacher's motets as a model. A comparative analysis of all three works provides further insight into compositional methods at the “dawn of the Baroque” era and sheds new light on what must have been a special teacher-student relationship.

New Directions in Duality Theory for Modal Logic

AbstractIn this work we present some new contributions towards two different directions in the study of modal logic. First we employ tense logics to provide a temporal interpretation of intuitionistic quantifiers as “always in the future” and “sometime in the past.” This is achieved by modifying the Gödel translation and resolves an asymmetry between the standard interpretation of intuitionistic quantifiers.Then we generalize the classic Gelfand–Naimark–Stone duality between compact Hausdorff spaces and uniformly complete bounded archimedean $\ell $ -algebras to a duality encompassing compact Hausdorff spaces with continuous relations. This leads to the notion of modal operators on bounded archimedean $\ell $ -algebras and in particular on rings of continuous real-valued functions on compact Hausdorff spaces. This new duality is also a generalization of the classic Jónsson-Tarski duality in modal logic.Abstract taken directly from the thesis.E-mail: lcarai@unisa.itURL: https://www.proquest.com/openview/5d284dbfb954383da9364149fa312b6f/1?pq-origsite=gscholar&cbl=18750&diss=y

A new type of intuitionistic fuzzy modal operators over intuitionistic fuzzy pairs

In the paper, two intuitionistic fuzzy modal operators from a new type are introduced over intuitionistic fuzzy pairs. Some of the basic properties of the new operators are formulated and checked.

Counterfactual epistemic scenarios

AbstractIn two‐dimensional semantics in the tradition of Davies and Humberstone, whether a singular term receives an epistemically shifted reading in the scope of a modal operator depends on whether the world considered as actual is shifted. This means that epistemically shifted readings should be available only in environments where an explicit contrast between the actual world and some counterfactual worlds cannot be made. In this paper, I argue that this is incorrect. Whether a singular term receives an epistemically shifted reading is independent of whether the world treated as actual is shifted. This, I argue, undermines the two‐dimensionalist account of epistemic shift. I then turn to the question how a positive view should handle these two phenomena separately. I argue for treating singular terms with a version of counterpart theory in which the difference between epistemically shifted and other readings is determined in the context of utterance.

Utilitarian attitude and norm in the discourse of Russian nationalists (on the material of the social network “VKontakte”)

Expressions with the predicative “expediently” manifest as a regular means of realization of modality of necessity, which is diverse in the semantic characteristics. The article aims to determine the type of the meaning of necessity and the nature of potentiality contained in such expressions in the discourse of Russian nationalists in the social network “VKontakte”. The article employs the methods semantic and pragmatic analysis. The author describes the predicative “expediently” as a modal operator of necessity, and the structure of the situation of necessity reflected by the expressions in the specified pragmatic conditions. The author reveals the semantic characteristics of combination of the utilitarian and deontic modalities that recognize such forms as the norm. It is established that expressions with the predicative “expediently” in the discourse of Russian nationalists manifest as the target norms and norms of operational preference. Such examples, due to their semantic and pragmatic characteristics, are the traditional method for realization of the speech act of appeal. The obtained results can be used in linguistic expertise, as they represent the significant diagnostic attributes of speech act of appeal in the pragmatic conditions of sociopolitical discourse. The scientific novelty is consists in the fact that this article is first to characterize the expressions with the predicative “expediently” as the forms of realization of meaning of the norm, which are recognized as the speech act of appeal in the discourse of Russian nationalists.

Constraining the identification of epistemic judges across different syntactic categories

As observed at various occasions, the usage of epistemic adverbs in information seeking questions is by far more restricted than the usage of epistemic adjectives. Starting from Lyons (1977) this contrast was motivated assuming that different types of epistemic operators come with different semantics and scope positions in the utterance, namely objective vs. subjective epistemic modality. However it is not possible to define clear classes of objective epistemic modal operators in terms of clear diagnostics. It will be shown here that the contrast of acceptability is more accurately explained in terms of locality and binding properties of the variable for the attitude holder rendering the epistemic judgement. If locally bound, epistemic modal operators can be embedded, if not, they are subject to much stricter conditions in order to be interpretable.

The grammatical representation of expletive negation

Expletive negation refers to constructions where a negator in the complement of certain lexical items does not change the polarity of the complement proposition. Jin & Koenig (2021) show that expletive negation occurs rather widely in languages of the world and in very similar environments. They propose a language production model of why such apparently illogical uses of negation arise in language after language. But their study does not address the grammatical status and representation of expletive negation. In this paper, we argue that expletive negation is part of the lexical knowledge speakers have of their language and that the negator in expletive negation constructions contributes a negation to a non-at-issue content associated with expletive negation triggers. We provide a Lexical Resource Semantics analysis of how triggers combine in a non-standard manner with the standard semantic content of their complements: the negation (and in some cases an additional modal operator) of the content of their complement is part of the trigger’s non-at-issue content while the scope of the negation is an argument of the trigger’s MAIN content. Finally, we suggest that the expletive use of the French negator ne includes a lexical constraint that requires it to modify a verb that reverse selects for an expletive negation trigger.

MODAL OPERATORS ON RINGS OF CONTINUOUS FUNCTIONS

AbstractIt is a classic result in modal logic, often referred to as Jónsson-Tarski duality, 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 for boolean algebras. Our goal is to 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 ${\sf KHaus}$ of compact Hausdorff spaces and the category $\boldsymbol {\mathit {uba}\ell }$ of uniformly complete bounded archimedean $\ell $ -algebras. We endow a bounded archimedean $\ell $ -algebra with a modal operator, which results in the category $\boldsymbol {\mathit {mba}\ell }$ of modal bounded archimedean $\ell $ -algebras. Our main result establishes a dual adjunction between $\boldsymbol {\mathit {mba}\ell }$ and the category ${\sf KHF}$ 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 ${\sf KHF}$ and the reflective subcategory $\boldsymbol {\mathit {muba}\ell }$ of $\boldsymbol {\mathit {mba}\ell }$ consisting of uniformly complete objects of $\boldsymbol {\mathit {mba}\ell }$ . This generalizes both Gelfand duality and Jónsson-Tarski duality.

Utilizing Treewidth for Quantitative Reasoning on Epistemic Logic Programs

AbstractExtending the popular answer set programming paradigm by introspective reasoning capacities has received increasing interest within the last years. Particular attention is given to the formalism of epistemic logic programs (ELPs) where standard rules are equipped with modal operators which allow to express conditions on literals for being known or possible, that is, contained in all or some answer sets, respectively. ELPs thus deliver multiple collections of answer sets, known as world views. Employing ELPs for reasoning problems so far has mainly been restricted to standard decision problems (complexity analysis) and enumeration (development of systems) of world views. In this paper, we take a next step and contribute to epistemic logic programming in two ways: First, we establish quantitative reasoning for ELPs, where the acceptance of a certain set of literals depends on the number (proportion) of world views that are compatible with the set. Second, we present a novel system that is capable of efficiently solving the underlying counting problems required to answer such quantitative reasoning problems. Our system exploits the graph-based measure treewidth and works by iteratively finding and refining (graph) abstractions of an ELP program. On top of these abstractions, we apply dynamic programming that is combined with utilizing existing search-based solvers like (e)clingo for hard combinatorial subproblems that appear during solving. It turns out that our approach is competitive with existing systems that were introduced recently.

Modal Logic S5 Satisfiability in Answer Set Programming

AbstractModal logic S5 has attracted significant attention and has led to several practical applications, owing to its simplified approach to dealing with nesting modal operators. Efficient implementations for evaluating satisfiability of S5 formulas commonly rely on Skolemisation to convert them into propositional logic formulas, essentially by introducing copies of propositional atoms for each set of interpretations (possible worlds). This approach is simple, but often results into large formulas that are too difficult to process, and therefore more parsimonious constructions are required. In this work, we propose to use Answer Set Programming for implementing such constructions, and in particular for identifying the propositional atoms that are relevant in every world by means of a reachability relation. The proposed encodings are designed to take advantage of other properties such as entailment relations of subformulas rooted by modal operators. An empirical assessment of the proposed encodings shows that the reachability relation is very effective and leads to comparable performance to a state-of-the-art S5 solver based on SAT, while entailment relations are possibly too expensive to reason about and may result in overhead.

Note on the Intuitionistic Logic of False Belief

In this paper we analyse logic of false belief in the intuitionistic setting. This logic, studied in its classical version by Steinsvold, Fan, Gilbert and Venturi, describes the following situation: a formula $\varphi$ is not satisfied in a given world, but we still believe in it (or we think that it should be accepted). Another interpretations are also possible: e.g. that we do not accept $\varphi$ but it is imposed on us by a kind of council or advisory board. From the mathematical point of view, the idea is expressed by an adequate form of modal operator $\mathsf{W}$ which is interpreted in relational frames with neighborhoods. We discuss monotonicity of forcing, soundness, completeness and several other issues. Finally, we mention the fact that it is possible to investigate intuitionistic logics of unknown truths.

Some results on certain properties of intuitionistic fuzzy sets

An operator is a special symbol for performing a specific function. Several operators like modal operators, topological operators, level operators, etc. have been defined over intuitionistic fuzzy sets. At the same time, so many operations were introduced and studied. The key objective of this paper is to study those operations over intuitionistic fuzzy sets and to investigate their properties. Some new results are obtained and proved.

Akiba's logic of indeterminacy

ABSTRACT A standard approach to indeterminacy treats ‘determinately’ and ‘indeterminately’ as modal operators. Determinacy behaves like necessity; indeterminacy like contingency. This raises two questions. What is the appropriate modal system for these operators? And how should we interpret that system? Ken Akiba has developed an account of ontic indeterminacy that interprets possible worlds as worldly precisifications. He argues that this account vindicates S4 as the logic of indeterminacy. In this paper I explore one significant and surprising consequence of this view. I do this in two stages. First, I prove a technical result, which I call the Infinite Indeterminacy Theorem. Roughly put, this theorem states that, at any given precisification w, either every instance of indeterminacy admits of never-ending higher-order indeterminacy, or else there's indeterminacy in an infinite amount of distinct atomic formulas. In slogan form: either all indeterminacy is infinitely ascending or else there's infinitely wide indeterminacy. Second, I unpack the metaphysical consequences of this technical result (under its intended interpretation) and assess their plausibility. I conclude that these consequences commit Akiba's theory to highly contentious – and arguably untenable – views about the metaphysics of indeterminacy and related matters.

The Vietoris functor and modal operators on rings of continuous functions

We introduce an endofunctor H on the category baℓ of bounded archimedean ℓ-algebras and show that there is a dual adjunction between the category Alg(H) of algebras for H and the category Coalg(V) of coalgebras for the Vietoris endofunctor V on the category of compact Hausdorff spaces. We prove that Gelfand duality lifts to a dual equivalence between Coalg(V) and the full reflective subcategory Algu(H) of Alg(H). We provide an alternate view of Algu(H) by introducing an endofunctor Hu on the full reflective subcategory of baℓ consisting of uniformly complete objects of baℓ and showing that Alg(Hu) is isomorphic to Algu(H). On the one hand, these results generalize those of [1,19] for the category of coalgebras of the Vietoris endofunctor on the category of Stone spaces. On the other hand, they provide an alternate, more categorical proof of a recent result of [6].

Doctrines, modalities and comonads

AbstractDoctrines are categorical structures very apt to study logics of different nature within a unified environment: the 2-categoryDtnof doctrines. Modal interior operators are characterised as particular adjoints in the 2-categoryDtn. We show that they can be constructed from comonads inDtnas well as from adjunctions in it, and we compare the two constructions. Finally we show the amount of information lost in the passage from a comonad, or from an adjunction, to the modal interior operator. The basis for the present work is provided by some seminal work of John Power.

Omitting types algebraically and more about amalgamation for modal cylindric algebras

AbstractLet α be an arbitrary infinite ordinal, and . In [26] we studied—using algebraic logic—interpolation and amalgamation for an extension of first order logic, call it , with α many variables, using a modal operator of a unimodal logic that contributes to the semantics. Our algebraic apparatus was the class of modal cylindric algebras. Modal cylindric algebras, briefly , are cylindric algebras of dimension α, expanded with unary modalities inheriting their semantics from a unimodal logic such as , or . When modal cylindric algebras based on are just cylindric algebras, that is to say, . This paper is a sequel to [26], where we study algebraically other properties of . We study completeness and omitting types (s) for s by proving several representability results for so‐called dimension complemented and locally finite . Furthermore, we study the notion of atom‐canonicity for , the variety of n‐dimensional modal cylindric algebras. Atom canonicity, a well known persistence property in modal logic, is studied in connection to for , which is restricted to the first n variables. We further continue our study of interpolation in [26] for algebraizable extensions of by studying using both algebraic logic and category theory. Our main results on are Theorems 3.7, 4.4 & 4.6, while our main results on amalgamation are Theorems 5.7, 5.10, 5.13 & 5.16.

Morphological, Structural, and Functional Networks Highlight the Role of the Cortical-Subcortical Circuit in Individuals With Subjective Cognitive Decline.

Subjective cognitive decline (SCD) is considered the earliest stage of the clinical manifestations of the continuous progression of Alzheimer’s Disease (AD). Previous studies have suggested that multimodal brain networks play an important role in the early diagnosis and mechanisms underlying SCD. However, most of the previous studies focused on a single modality, and lacked correlation analysis between different modal biomarkers and brain regions. In order to further explore the specific characteristic of the multimodal brain networks in the stage of SCD, 22 individuals with SCD and 20 matched healthy controls (HCs) were recruited in the present study. We constructed the individual morphological, structural and functional brain networks based on 3D-T1 structural magnetic resonance imaging (sMRI), diffusion tensor imaging (DTI) and resting-state functional magnetic resonance imaging (rs-fMRI), respectively. A t-test was used to select the connections with significant difference, and a multi-kernel support vector machine (MK-SVM) was applied to combine the selected multimodal connections to distinguish SCD from HCs. Moreover, we further identified the consensus connections of brain networks as the most discriminative features to explore the pathological mechanisms and potential biomarkers associated with SCD. Our results shown that the combination of three modal connections using MK-SVM achieved the best classification performance, with an accuracy of 92.68%, sensitivity of 95.00%, and specificity of 90.48%. Furthermore, the consensus connections and hub nodes based on the morphological, structural, and functional networks identified in our study exhibited abnormal cortical-subcortical connections in individuals with SCD. In addition, the functional networks presented more discriminative connections and hubs in the cortical-subcortical regions, and were found to perform better in distinguishing SCD from HCs. Therefore, our findings highlight the role of the cortical-subcortical circuit in individuals with SCD from the perspective of a multimodal brain network, providing potential biomarkers for the diagnosis and prediction of the preclinical stage of AD.

On the Operation ∆ over Intuitionistic Fuzzy Sets

Recently, the new operation ∆ was introduced over intuitionistic fuzzy sets and some of its properties were studied. Here, new additional properties of this operations are formulated and checked, providing an analogue to the De Morgan’s Law (Theorem 1), an analogue of the Fixed Point Theorem (Theorem 2), the connections between the operation ∆ on one hand and the classical modal operators over IFS Necessity and Possibility, on the other (Theorems 3 and 4). It is shown that it can be used for a de-i-fuzzification. A geometrical interpretation of the process of constructing the operator ∆ is given.

Efficient loop-check for multimodal KD45n logic

We introduce sequent calculus for multi-modal logic KD45n which uses efficient loop-check. Efficiency of the used loop-check is obtained by using marked modal operator squarei which is used as an alternative to sequent with histories ([2,3]).We use inference rules with or branches to make all rules invertible or semi-invertible. We showthe maximum height of the constructed derivation tree. Also polynomial space complexity is proved.

Modal systems for covering semantics and boundary operator

Towards the study of covering based rough set semantics for modal logic [39], we present a modal system corresponding to the covering systems P3,C1, and CGr. We also study the modal systems for boundary operators based on generalized approximation spaces as well as covering systems. Our study also leads to covering semantics for some contingency logics and provide its connection with rough set theory. Further, an alternative modal system based on the contingency modal operator is provided corresponding to the covering system P4.