Several remarks on generating a theoretical model of modal logic

Electronic Notes in Theoretical Computer Science, 2014

We introduce a sound and complete graph calculus for multi-modal logics. This formalism internalizes the (Kripke) semantics of modal logics and provides uniform tools for expressing and manipulating modal formulas. We present the graph calculus for logic K and show how to extend it to handle some modalities, like the global and difference modalities, in a natural manner. We also indicate how it can be easily extended to other normal modal logics, such as T, S4, S5, etc.

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Studies in Logic and Practical Reasoning, 2007

This chapter introduces modal logic as a tool for talking about graphs, or to use more traditional terminology, as a tool for talking about Kripke models and frames. We want the reader to gain an intuitive appreciation of this perspective, and a firm grasp of the key technical ideas (such as bisimulations) which underly it. We introduce the syntax and semantics of basic modal logic, discuss its expressivity at the level of models, examine its computational properties, and then consider what it can say at the level of frames. We then move beyond the basic modal language, examine the kinds of expressivity offered by a number of richer modal logics, and try to pin down what it is that makes them all 'modal'. We conclude by discussing an example which brings many of the ideas we discuss into play: games. Contents 1 Introduction 2 Basic modal logic 2.1 First steps in relational semantics 2.2 The standard translation 3 Simulation and definability 3.1 Drawing distinctions 3.2 Structural invariances: bisimulation 3.3 Invariance and definability in first-order logic 3.4 Invariance and definability in modal logic 3.5 Modal logic and first-order logic compared 3.6 Bisimulation as a game 4 Computation and complexity 4.1 Model checking 4.2 Decidability 4.3 Complexity 4.4 Other reasoning tasks 5 Richer logics 5.1 Axioms and relational frame properties 5.2 Frame correspondence and second-order logic 5.3 First-order definable modal axioms 5.4 Correspondence in richer languages: fixed-point extensions 5.5 Modally definable frame classes 5.6 First-order logic as modal logic 6 Richer languages 6.1 The universal modality 6.2 Hybrid logic 6.3 Temporal logic with Until and Since operators 6.4 Conditional logic 6.5 The guarded fragment 38 6.6 Propositional Dynamic Logic 40 6.7 Modal µ-calculus 42 6.8 General perspectives 45 7 New descriptive challenges 46 7.1 An example where it all comes together: games 46 References 48

Studies in Logic and Practical Reasoning, 2007

In Proceedings of the 7th annual ACM symposium on …, 1985

In Kripke semantics for modal logic, "possible world^" and the possibility relation are both primitive notions. This has both technical and conceptual shortcomings. From a technical point of view, the mathematics associated with Kripke. semantics is often quite complicated. From a conceptual point of view, it is not clear how to use Kripke structures to model know!edge and belief, where one wants a clearer understanding of the notions that are primitive in Kripke semantics. We introduce modal structures as models for modal logic. We use the idea of possible worlds, but by directly describing the "internal semantics" of each possible world. It is much easier to study the standard logical questions, such as completeness, decidability, and compactness, ushg modal structures. Furthermore, modal structures offer a much more intuitive approach to modelling knowledge and belief.

Artificial Intelligence in Medicine, 1997

Book reviews Sally Popkorn. First Steps in Modal Logic, Cambridge University Press, Cambridge, 1994, 314 pp., ISBN 052146482, (hardback) f25.00, US$39.95.

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1984

Zeilsclrr. 1. muth. L w i k und Grundloqen d . blalh. Models of the language are systems of the form M = (Ob, At, {ind(A)}o+Acr,, m) defined as in section 5. We define satisfiability of formulas by non-empty subsets of set Ob. We say that set K satisfies formula A in model M (K sat, A ) whenever the following conditions are hsatisfied :

2007

In [H1], we develop a fixpoint semantics, the least model semantics, and an SLDresolution calculus in a direct way for positive modal logic programs in basic serial monomodal logics. In [H2], we extend the results and generalize the methods of [H1] for multimodal logics, giving a framework for developing semantics for positive logic programs in serial multimodal logics. In [H3], we report on our design and implementation of the modal logic programming system [18] based on the framework given in [H2]. The subject of modal deductive databases closely relates to modal logic programming. In [H5], using our theory of modal logic programming [H1, H2], we formulate modal deductive databases and give computational methods for the modal query language MDatalog. In [H5], we also estimate data complexity of MDatalog in some multimodal logics. The work [H4] contains our results on data complexity of MDatalog in basic monomodal logics.

Advances in Modal Logic, 2024

Non-classical generalizations of classical modal logic have been developed in the contexts of constructive mathematics and natural language semantics. In this paper, we discuss a general approach to the semantics of non-classical modal logics via algebraic representation theorems. We begin with complete lattices L equipped with an antitone operation ¬ sending 1 to 0, a completely multiplicative operation □, and a completely additive operation ◊. Such lattice expansions can be represented by means of a set X together with binary relations ◁, R, and Q, satisfying some first-order conditions, used to represent (L, ¬), □, and ◊, respectively. Indeed, any lattice L equipped with such a ¬, a multiplicative □, and an additive ◊ embeds into the lattice of propositions of a frame (X, ◁, R, Q). Building on our recent study of fundamental logic, we focus on the case where ¬ is dually self-adjoint (a ≤ ¬b implies b ≤ ¬a) and ◊¬a ≤ ¬□a. In this case, the representations can be constrained so that R = Q, i.e., we need only add a single relation to (X, ◁) to represent both □ and ◊. Using these results, we prove that a system of fundamental modal logic is sound and complete with respect to an elementary class of bi-relational structures (X, ◁, R).

Trans Roy Soc Trop Med Hyg, 2010

Modal logic was born in the early part of the 20th century as a branch of logic applied to the analysis of philosophical notions and issues. While it still retains a bit of this grandeur, today, modal logic sits at a crossroads of many academic disciplines, and thus, it provides a unique vantage point for students with broad interdisciplinary interests. These notes are the accumulated material for a course taught for many years at Stanford to students in philosophy, symbolic systems, linguistics, computer science, and other fields. The purpose is to give them a modern introduction to modal logic, beyond lingering conceptions dating back to the distant past -and topics include both technical perspectives, and a wide range of applications showing the current range of the field. To check if the picture in these notes is representative, the reader may consult the 2006 Handbook of Modal Logic, Elsevier, Amsterdam, co-edited with my colleagues Patrick Blackburn and Frank Wolter, Elsevier Science Publications, Amsterdam. For philosophers, it may also be of interest to check with my 1988 lecture notes Manual of Intensional Logic, CSLI Publications, Stanford, which then represented my ideal of a modern introduction to the field. Some topics have panned out, but others have proved remarkably wide off the mark.