Modal operator

About: Modal operator is a research topic. Over the lifetime, 1151 publications have been published within this topic receiving 22865 citations. The topic is also known as: modal connective.

Dynamic Sequent Calculus for the Logic of Epistemic Actions and Knowledge

Abstract: Dynamic Logics (DLs) form a large family of nonclassical logics, and perhaps the one enjoying the widest range of applications. Indeed, they are designed to formalize change caused by actions of diverse nature: updates on the memory state of a computer, displacements of moving robots in an environment, measurements in models of quantum physics, belief revisions, knowledge updates, etc. In each of these areas, DL-formulas express properties of the model encoding the present state of affairs, as well as the preand post-conditions of a given action. Actions are semantically represented as transformations of one model into another, encoding the state of affairs after the action has taken place. DL-languages are expansions of classical (static) logic with dynamic operators, parametrized with actions; dynamic operators are modalities interpreted in terms of the transformation of models corresponding to their action-parameters. However, when dynamic logics feature both dynamic and ‘static’ modalities, as in the case of the Dynamic Epistemic Logics (DEL), they typically lose many desirable properties, such as the closure under uniform substitution, which facilitate a smooth algebraic and proof-theoretic treatment. This and other difficulties make their algebraic and proof-theoretic theory presently underdeveloped, and the few existing proposals seem quite ad hoc: in particular, no proposed calculus in the literature enjoys the minimal conditions enabling the specification of the meaning of a logical symbol in the sense of proof-theoretic semantics. We developed a family of display-style, cut-free sequent calculi for dynamic epistemic logics on both an intuitionistic and a classical base [7]. Like the standard display calculi, these calculi are modular: just by modifying the structural rules according to Dosen’s principle [12], these calculi are generalizable both to different Dynamic Logics (Epistemic, Deontic, etc.) and to different propositional bases (Linear, Relevant, etc.). Moreover, the rules they feature agree with the standard relational semantics for dynamic epistemic logics, while a ‘natural’ coalgebraic semantics was introduced in [7]. However, these calculi still feature a peculiar set of contextual rules, i.e. structural and operational rules involving a context (understood as the precondition for the applicability of an action), which make them non-standard as display calculi. To fix this shortcoming, in [8] we propose a new class of more general sequent calculi (which we call dynamic calculi) that overcome the need of contextual rules and enjoy other benefits, including a natural proof-theoretic semantics. In this talk we present two concrete examples of dynamic calculi for one specific dynamic epistemic logic [2], both in its classical and in its intuitionistic version, where the Intuitionistic logic for Epistemic Action and Knowledge was intoduced via duality in [10] defining dynamic algebraic semantics on intuitionistic base. These calculi can be understood as generalizations of the display calculi [4], since they feature: 1. constituents of several types (e.g. of type AG for ‘agents’ or ACT for ‘actions’), along with propositional constituents, which in their turn can be of different types (e.g. intuitionistic propositions iPROP, or classical propositions cPROP, etc.); N. Galatos, A. Kurz, C. Tsinakis (eds.), TACL 2013 (EPiC Series, vol. 25), pp. 85–87 85 Sequent calculus for dynamic epistemic logic Greco, Kurz and Palmigiano 2. non-contextual1 structural connectives (structural conjunction and disjunction, implication and disimplication — also called exclusion —, and their neutral elements) by which structures are built, where a structure can be homogeneous (e.g. α ∧ β is of type ACT ×ACT ) or heterogeneous (e.g. α ∧ A is of type ACT × PROP).2 3. structural introduction rules, which constitues a pure structural calculus because the structural and operational level are now completely separated (e.g. (X ` Y)∧ (W ` Z) / X∧W ` Y∧ Z); 4. unary translation rules, to pass from structural to operational level: the flattening rule, making it possible to transform homogeneous structures into operational formulas (e.g. A ∧ B ` X / A∧B ` X, where A∧B is a propositional formula of type PROP×PROP→ PROP), and the currying rule, making it possible to transform heterogeneous structures into ‘parametrized’ operational formulas (e.g. α ∧ A ` X / 〈α〉A ` X, where 〈α〉A is a propositional modal formula of type ACT ×PROP→ PROP). These calculi are interesting from both a methodological and an applicative viewpoint. Methodologically, the dynamic calculi achieve a unified framework simultaneously accounting for different logical behaviors: modalities (e.g. exponentials or temporal operators) and quantifiers can be seen in terms of combinations of heterogeneous components by means of operational rules at the merging-level; the behavior of a single modal operator (e.g. monotonicity) or the interaction between modalities (e.g. dynamic and epistemic operators) can be captured by means of structural rules at the merging-level. We will also discuss the categorial semantics, which seems the most ‘appropriate’ for this kind of calculi. From the point of view of the applications, dynamic calculi provide a way of generating specific display calculi for a wide array of logics — which includes but is not limited to dynamic (epistemic) logics — simply by deriving the appropriate rules; this allows to obtain meta-theorems such as cut elimination [8] uniformly for each specific calculus as a byproduct of the general theory of dynamic calculi. The dynamic calculi enjoy the strong version of Wansing’s fundamental properties of segregation, symmetricity and explicitness [12]. Therefore, they share all the benefits that come from the display calculi, but are more general: for instance they capture also dynamic poly-modal logics on substructural base.

Проблематика модальных глаголов в современном китайском языке

Abstract: The article presents a new view on Chinese modal verbs as a part of speech. Based on the typology of the Chinese language, the authors analyzed modal verbs according to their functional-syntactic, formal-morphological, and semantic features in order of importance. The article discusses the position of modal operators in the sentence and other characteristics. For instance, Chinese modal verbs have no impact on the object and cannot independently form sentences or combine with grammemes. Therefore, the authors believe that Chinese modal verbs (modal operators) belong to the lexical-grammatical group of adverbs as a special category of intentional adverbs. Their intentionality reflects the outward focus of linguistic consciousness, based on the internal reference point of the speaker. The group includes such meanings as "wish", "obligation", "opportunity", "permission", and "will". The research owes its theoretical significance to the fact that it contributes to a better understanding of the essence and nature of modal operators and modality meanings, identifying them as a separate group of adverbs. The obtained results are applicable in the field of theoretical grammar of the Chinese language and can be used in researches related to further analysis of parts of speech problem in the Chinese language.

Two types of dispositional adjectives 1

Abstract: This paper investigates a class of adjectives in Brazilian Portuguese formed with the suffix -vel. I will refer to them as dispositional adjectives. An interesting property of these adjectives is related to their different semantic interpretations, specifically with the (un)availability of a possibility modal reading. The aim of this paper is to provide a better understanding of what sets these two classes apart. I propose to analyze the suffix -vel as a type of dynamic modal operator relativized to a modal base comprised of properties, not propositions, following Brennan (1993). I argue that the nature of the properties in question, subjective or objective, and the manifestation of dispositions are the source of the aforementioned semantic contrasts in modality, which are also related to eventivity and stativity.

A note on Gabriel Uzquiano's 'Varieties of Indefinite Extensibility'

Abstract: Gabriel Uzquiano has offered an account of indefinite extensibility for sets in the context of a modal logic. The modal operators are interpreted in terms of linguistic extensibility. After reviewing the proposal, I argue that the view should be understood as a version of in rebus structuralism about set theory. As such it is subject to the usual problems for in rebus structuralism. In particular, there is no good extra set-theoretic reason to assent to an ontology of sufficient cardinality to make true the theorems of ZFC.

A Conservative Approach to Distributed Belief Fusion

Abstract: In this paper, we develop logics for merging beliefs of agents with different degrees of reliability. The logics are obtained by combining the multiagent epistemic logic from [8] and multi-sources reasoning systems from [2]. Every ordering for the reliability of the agents is represented by a modal operator, so we can reason with the merging information under different situations. The approach is conservative in the sense that if an agent’s belief is in conflict with those of higher priorities, then his belief is completely discarded from the merged result. We consider two strategies for the conservative merging of beliefs. In the first one, if inconsistency occurs at some level, then all beliefs at the lower levels are discarded simultaneously, so it is called level cutting strategy. For the second one, only the level at which the inconsistency occurs is skipped, so it is called level skipping strategy. The formal semantics and axiomatic systems for these two strategies are presented.