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.

A Symmetric Approach to Axiomatizing Quantifiers and Modalities

Abstract: We present an axiomatization of several of the basic modal logics, with the idea of giving the two modal operators □ and ◊ equal weight as far as possible. Then we present a parallel axiomatization of classical quantification theory, working our way up through a sequence of rather curious subsystems. It will be clear at the end that the essential difference between quantifiers and modalities is amusing in a vacuous sort of way. Finally we sketch tableau proof systems for the various logics we have introduced along the way. Also, the “natural” model theory for the subsystems of quantification theory that come up is somewhat curious. In a sense, it amounts to a “stretching out” of the Henkin-style completeness proof, severing the maximal consistent part of the construction quite thoroughly from the part of the construction that takes care of existential-quantifier instances.

An Interval-Based Modal Logic for System Specification

Abstract: This paper presents an interval-based modal logic for specification of real-time systems and knowledge representation. A real-time system is understood as a time-dependent dynamic entity whose attributes can be observed and changed by performing actions. We take linear and dense time as our underlying time model, which can be characterised by the rationals. In our formalism, modal operators on formulas are defined by time intervals and modal operators on terms are defined by time instants and actions. Actions are not instantaneous but take some time to be performed. Different actions can take place at the same time. The proposed logic can also allow to specify obligations and prohibitions of actions over a period of time in deontic sense.

Modal Operators and Functional Completeness, II

Abstract: In the Kripke semantics for propositional modal logic, a frame W = (W, <) represents a set of "possible worlds" and a relation of "accessibility" between possible worlds. With respect to a fixed frame IW, a proposition is represented by a subset of W (regarded as the set of worlds in which the proposition is true), and an n-ary connective (i.e. a waybf forming a new proposition from an ordered n-tuple of given propositions) is represented by a function fW: (p(W))y -* P(W). Finally a state of affairs (i.e. a consistent specification whether or not each proposition obtains) is represented by an ultrafilter over W. {To avoid possible confusion, the reader should forget that some people prefer the term "states of affairs" for our "possible worlds".} In a broader sense, an n-ary connective is represented by an n-ary operator f = {fw 'I E Fr}, where Fr is the class of all frames and each fW (P(W))n -* P(W). A connective is modal if it corresponds to a formula of propositional modal logic. A connective C is coherent if whether C(Pi, * *, Pn) is true in a possible world depends only upon which modal combinations of P1, * * , Pn are true in that world. (A modal combination of P1, * * *, Pn is the result of applying a modal connective to P1, * * , Pn.) A connective C is strongly coherent if whether C(P1, * * , Pn) obtains in a state of affairs depends only upon which modal combinations of P1, * * *, Pn obtain in that state of affairs. In ?1 we study the classes of modal, coherent, and strongly coherent connectives, via the corresponding classes of operators. We shall characterize model-theoretically (i.e. without reference to the formulas of modal logic) these classes of operators, and we shall see that "modal" and "strongly coherent" are equivalent, and that they imply "coherent" but not conversely. A (normal modal propositional) logic L is functionally complete if every coherent operator on Fr(L), the class of all frames for L, is modal. The usual functional completeness theorems for the classical propositional calculus (every truth table is realized by a formula) and for S5 (every array of partial truth tables is realized by a formula [3, ?38.0]) may be interpreted as asserting the functional completeness, in the present sense, of CPC (i.e. K + (p <-+ Lp)) and of S5. In ?2 we shall determine which logics are functionally complete.

A Logical Description of Metaphor Analysis

Abstract: This paper aims to use logical techniques to describe how metaphors are analyzed. Metaphor analysis process functions as one of the most important strategies to uncover implied information in discourse understanding. A metaphor analysis logic system is developed and presented in terms of its definitions, axiomatic system, inference rules, properties, semantic interpretations and applications. The merits of the logic are that possible worlds are substituted with possible feature spaces compared with Local Frame Theory, and an understanding modal operator Up, a relational symbol < and a Gestalt rule are embodied. The most notable feature of the logic is that it takes into account subjective factors in the process of metaphor analysis.