Interpretation Of Modal Logic Research Articles

Geometric Model Checking of Continuous Space

Topological Spatial Model Checking is a recent paradigm where model checking techniques are developed for the topological interpretation of Modal Logic. The Spatial Logic of Closure Spaces, SLCS, extends Modal Logic with reachability connectives that, in turn, can be used for expressing interesting spatial properties, such as "being near to" or "being surrounded by". SLCS constitutes the kernel of a solid logical framework for reasoning about discrete space, such as graphs and digital images, interpreted as quasi discrete closure spaces. Following a recently developed geometric semantics of Modal Logic, we propose an interpretation of SLCS in continuous space, admitting a geometric spatial model checking procedure, by resorting to models based on polyhedra. Such representations of space are increasingly relevant in many domains of application, due to recent developments of 3D scanning and visualisation techniques that exploit mesh processing. We introduce PolyLogicA, a geometric spatial model checker for SLCS formulas on polyhedra and demonstrate feasibility of our approach on two 3D polyhedral models of realistic size. Finally, we introduce a geometric definition of bisimilarity, proving that it characterises logical equivalence.

On the Sick Father, the Repentant Sinner, and Other Problems in Medieval Deontic Logic

AbstractMany medieval philosophers and logicians regarded modal logic as a model for epistemic, normative and optative concepts on the basis of their apparent similarities to modal concepts. These interpretations of modal logic led to critical studies of the applicability of modal principles to deontic concepts. Some philosophers presented apparent counterexamples to the deontic interpretation of the Consequence Principle, according to which the consequences of what is obligatory (or permitted) are also obligatory (permitted). These examples are variants of Lennart Åqvist's paradox of the Good Samaritan. In this article it is argued that the examples involve a confusion between the compounded and the divided senses of deontic propositions, and are not genuine counterexamples to the Consequence Principle. If the concept of knowledge is analysed as epistemically satisfactory belief, the paradox of epistemic obligation can be solved in a similar way.

Spatial logic of tangled closure operators and modal mu-calculus

There has been renewed interest in recent years in McKinsey and Tarski's interpretation of modal logic in topological spaces and their proof that S4 is the logic of any separable dense-in-itself metric space. Here we extend this work to the modal mu-calculus and to a logic of tangled closure operators that was developed by Fernández-Duque after these two languages had been shown by Dawar and Otto to have the same expressive power over finite transitive Kripke models. We prove that this equivalence remains true over topological spaces.We extend the McKinsey–Tarski topological ‘dissection lemma’. We also take advantage of the fact (proved by us elsewhere) that various tangled closure logics with and without the universal modality ∀ have the finite model property in Kripke semantics. These results are used to construct a representation map (also called a d-p-morphism) from any dense-in-itself metric space X onto any finite connected locally connected serial transitive Kripke frame.This yields completeness theorems over X for a number of languages: (i) the modal mu-calculus with the closure operator ◇; (ii) ◇ and the tangled closure operators 〈t〉 (in fact 〈t〉 can express ◇); (iii) ◇,∀; (iv) ◇,∀,〈t〉; (v) the derivative operator 〈d〉; (vi) 〈d〉 and the associated tangled closure operators 〈dt〉; (vii) 〈d〉,∀; (viii) 〈d〉,∀,〈dt〉. Soundness also holds, if: (a) for languages with ∀, X is connected; (b) for languages with 〈d〉, X validates the well-known axiom G1. For countable languages without ∀, we prove strong completeness. We also show that in the presence of ∀, strong completeness fails if X is compact and locally connected.

Refinement Quantified Logics of Knowledge

Refinement quantifiers were introduced to quantify over all refinements of a model in modal logic, where a refinement is described by a simulation relation. Given the “possible worlds” interpretation of modal logic, a refinement corresponds to an agent ruling out certain possible worlds based on new information. Recent work by van Ditmarsch, French and Pinchinat has presented an axiomatization and complexity results for refinement quantifiers in the general modal logic K. Here we extend these results to apply to the epistemic and doxastic settings for a single agent.

Modal Logics of Reactive Frames

In this chapter the author explains modal logics of reactive frames. He shows the correspondence between logic, reactive frame property, and the shattered frame unfolded property. He uses classical tools to study completeness in a new interpretation of modal logic, successfully studying some of its subsystems.

An algebraic generalization of Kripke structures

AbstractThe Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4 and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL and the ramified temporal logic CTL.

A survey of modal logics characterising behavioural equivalences for non-deterministic and stochastic systems

Behavioural equivalences are a means of establishing whether computing systems possess the same properties. The specific set of properties that are preserved by a specific behavioural equivalence clearly depends on how the system behaviour is observed and can usually be characterised by means of a modal logic. In this paper we consider three different approaches to the definition of behavioural equivalences – bisimulation, testing and trace – applied to three different classes of systems – non-deterministic, probabilistic and Markovian – and we survey the nine resulting modal logic characterisations, each of which stems from the Hennessy–Milner logic. We then compare the nine characterisations with respect to the logical operators, in order to emphasise the differences between the three approaches in the definition of behavioural equivalences and the regularities within each of them. In the probabilistic and Markovian cases we also address the issue of whether the probabilistic and temporal aspects should be treated in a local or global way and consequently whether the modal logic interpretation should be qualitative or quantitative.

The modal logic of forcing

A set theoretical assertionψ\psiisforceableorpossible, written◊ψ\Diamond \psi, ifψ\psiholds in some forcing extension, andnecessary, written◻ψ\Box \psi, ifψ\psiholds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if ZFC is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theoryS4.2\mathsf {S4.2}.

Understanding the Brandenburger-Keisler Paradox

Adam Brandenburger and H. Jerome Keisler have recently discovered a two person Russell-style paradox. They show that the following configurations of beliefs is impossible: Ann believes that Bob assumes that Ann believes that Bob’s assumption is wrong. In [7] a modal logic interpretation of this paradox is proposed. The idea is to introduce two modal operators intended to represent the agents’ beliefs and assumptions. The goal of this paper is to take this analysis further and study this paradox from the point of view of a modal logician. In particular, we show that the paradox can be seen as a theorem of an appropriate hybrid logic.

Fuzzy Truthoods Based on an Additive Semantic Measure with Break of Global Symmetry in Modal Logic

We propose an additive ”semantic measure” theory based on the model of a perception based fuzzy truthoods in analogy to probability measure. Probability measure, which is the subject of the probability theory, is a special case of the fuzzy measure in the Dempster-Shafer theory when the focal sets are singletons. In the literature, the fuzzy measure is considered an extension of the probability measure. In the Dempster theory, the imprecise probabilities are defined with AND, OR and NOT operations to determine the lower and the upper probabilities. Currently, in the fuzzy set and the fuzzy logic theory, there is no analogous structure or connection to the probability theory. In this paper, we show that there exist a connection between the probability theory and the fuzzy set theory via its operations. We show that when we move from the context of the events to the context of possible worlds, in the modal logic, an isomorphism exist between the probability theory and the meta-theory based upon modal logic. With this isomorphism, it is possible to use the rules in the probability theory to formulate, in the context of possible worlds, the fuzzy logic rules with AND, OR and NOT operation with the t-norms, t-conorms and the fuzzy complement. When the formal theory of the probability is used for sets of worlds, the meaning of the probability changes dramatically. In the ordinary probability theory every event is separate, i.e., mutually exclusive, from the other events and is logically well defined. In the context of possible worlds, the events are semantic events that are not mutually exclusive, for which agents or experts assign different logic values in different worlds (contexts) to the same object. Since it is impossible to identify all the properties of an object, we evaluate hypothetical properties of the object from a partially open set of data or point of view. For this purpose the unknown property space of an entity or an object is divided into parts or granules, i.e., worlds, where hypothetical properties can be evaluated. However in the hypothetical nature of the local or context dependent evaluations, there are the possibilities of conflicting evaluations. That is the property evaluations of an object are substituted with a set of evaluations embedded in a conceptual space where granules are worlds. In this context, we propose an additive semantic measure theory that is based on the model of the fuzzy truthoods. Transformations in the logic values of a set of worlds generate asymmetry in the modal logic interpretation of uncertainty. At every transformation we associate a break of symmetry that we can compensate to generate new modus ponens and new tautology in the fuzzy logic. Similarities with the gauge theory in physics are possible.

The Knower Paradox in the Light of Provability Interpretations of Modal Logic

The Knower Paradox in the Light of Provability Interpretations of Modal Logic

The Knower Paradox in the light of provability interpretations of modal logic

This paper propounds a systematic examination of the link between the Knower Paradox and provability interpretations of modal logic. The aim of the paper is threefold: to give a streamlined presentation of the Knower Paradox and related results; to clarify the notion of a syntactical treatment of modalities; finally, to discuss the kind of solution that modal provability logic provides to the Paradox. I discuss the respective strength of different versions of the Knower Paradox, both in the framework of first-order arithmetic and in that of modal logic with fixed point operators. It is shown that the notion of a syntactical treatment of modalities is ambiguous between a self-referential treatment and a metalinguistic treatment of modalities, and that these two notions are independent. I survey and compare the provability interpretations of modality respectively given by Skyrms, B. (1978, The Journal of Philosophy 75: 368--387) Anderson, C.A. (1983, The Journal of Philosophy 80: 338-- 355) and Solovay, R. (1976, Israel Journal of Mathematics 25: 287--304). I examine how these interpretations enable us to bypass the limitations imposed by the Knower Paradox while preserving the laws of classical logic, each time by appeal to a distinct form of hierarchy.

Reasoning About Space: The Modal Way

We investigate the topological interpretation of modal logic in modern terms, using a new notion of bisimulation. We look at modal logics with interesting topological content, presenting, among others, a new proof of McKinsey and Tarski's theorem on completeness of S4 with respect to the real line, and a completeness proof for the logic of finite unions of convex sets of reals. We conclude with a broader picture of extended modal languages of space, for which the main logical questions are still wide open.

Mathematical modal logic: A view of its evolution

This is a survey of the origins of mathematical interpretations of modal logics, and their development over the last century or so. It focuses on the interconnections between algebraic semantics using Boolean algebras with operators and relational semantics using structures often called Kripke models. It reviews the ideas of a number of people who independently contributed to the emergence of relational semantics, and compares them with the work of Kripke. It concludes with an account of several applications of modal model theory to mathematics and theoretical computer science.

Evidence theory in multivalued models of modal logic

ABSTRACT A modal logic interpretation of Dempster-Shafer theory is developed in the framework of multivalued models of modal logic, i.e. models in which in any possible world an arbitrary number (possibly zero) of atomic propositions can be true. Several approaches to conditioning in multivalued models of modal logic are presented.

Dempster–Shafer theory framed in modal logic

In this paper, the modal logic interpretation of plausibility and belief measures on an arbitrary universe of discourse, as proposed by Harmanec et al., is further developed by employing notions from set-valued analysis. In a model of modal logic, a multivalued mapping is constructed from the accessibility relation and a mapping determined by the value assignment function. This multivalued mapping induces a plausibility measure and a belief measure expressed in terms of conditional probabilities of inverse and superinverse images, or equivalently, in terms of conditional probabilities of truth sets of possibilitations and necessitations. Restricting to a finite universe of discourse, multivalued interpretations of basic probability assignments and of commonality functions are also obtained, in terms of conditional probabilities of pure inverse and subinverse images, or equivalently, in terms of conditional probabilities of truth sets of particular logical expressions involving possibilitations and necessitations.

Dempster's rule of conditioning translated into modal logic

A modal logic interpretation of Dempster's rule of conditioning is developed. It is shown that by restricting a model of modal logic in a non-trivial way, the measures induced by this restricted model are, in fact, the conditional measures given the restricting set, corresponding to the measures induced by the original model.

CONCEPTUAL FOUNDATIONS OF QUANTUM MECHANICS: THE ROLE OF EVIDENCE THEORY, QUANTUM SETS, AND MODAL LOGIC

Recognizing that syntactic and semantic structures of classical logic are not sufficient to understand the meaning of quantum phenomena, we propose in this paper a new interpretation of quantum mechanics based on evidence theory. The connection between these two theories is obtained through a new language, quantum set theory, built on a suggestion by J. Bell. Further, we give a modal logic interpretation of quantum mechanics and quantum set theory by using Kripke's semantics of modal logic based on the concept of possible worlds. This is grounded on previous work of a number of researchers (Resconi, Klir, Harmanec) who showed how to represent evidence theory and other uncertainty theories in terms of modal logic. Moreover, we also propose a reformulation of the many-worlds interpretation of quantum mechanics in terms of Kripke's semantics. We thus show how three different theories — quantum mechanics, evidence theory, and modal logic — are interrelated. This opens, on one hand, the way to new applications of quantum mechanics within domains different from the traditional ones, and, on the other hand, the possibility of building new generalizations of quantum mechanics itself.

Modal logic interpretation of Dempster-Shafer theory: An infinite case

A modal logic interpretation of belief and plausibility measures defined on infinite sets is established. As a special case, a modal logic interpretation of necessity and possibility measures defined on infinite sets is also established. It is proven in both cases that the interpretation is complete. These results establish, in effect, modal logic interpretations of the Dempster-Shafer theory and possibility theory.

On modal logic interpretation of Dempster–Shafer theory of evidence

This article further develops one branch of research initiated in an article by Resconi, Klir, and St. Clair (G. Resconi, G. J. Klir, and U. St. Clair, Int. J. Gen. Syst., 21(1), 23-50 (1992) and continued in another article by Resconi et al. (Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems, 1(1), 1993). It fully formulates an interpretation of the Dempster-Shafer theory in terms of the standard semantics of modal logic. It is shown how to represent the basic probability assignment function as well as the commonality function of the Dempster-Shafer theory by modal logic and that this representation is complete for rational-valued functions (basic assignment, belief, or plausibility functions). © 1994 John Wiley & Sons, Inc.