Showing papers on "Normal modal logic published in 1998"

First-Order Modal Logic

Abstract: Preface. 1. Propositional Modal Logic. 2. Tableau Proof Systems. 3. Axiom Systems. 4. Quantified Modal Logic. 5. First-Order Tableaus. 6. First-Order Axiom Systems. 7. Equality. 8. Existence and Actualist Quantification. 9. Terms and Predicate Abstraction. 10. Abstraction Continued. 11. Designation. 12. Definite Descriptions. References. Index.

Products of modal logics, part 1

Abstract: The paper studies many-dimensional modal logics corresponding to products of Kripke frames. It proves results on axiomatisability, the finite model property and decidability for product logics, by applying a rather elaborated modal logic technique: p-morphisms, the finite depth method, normal forms, filtrations. Applications to first order predicate logics are considered too. The introduction and the conclusion contain a discussion of many related results and open problems in the area.

Displaying Modal Logic

Abstract: Preface. 1. Introduction. 2. Sequents Generalized. 3. Display Logic. 4. Properly Displayable Logics, Displayable Logics and Strong Cut-Elimination. 5. A Proof-Theoretic Proof of Functional Completeness for Many Modal and Tense Logics. 6. Modal Tableaux Based on Residuation. 7. Strong Cut-Elimination and Labelled Modal Tableaux. 8. Tarskian Structured Consequence Relations and Functional Completeness. 9. Constructive Negation and the Modal Logic of Consistency. 10. Displaying as Temporalizing. 11. Translation of Hypersequents into Display Sequents. 12. Predicate Logics on Display. 13. Appendix. Bibliography. Index.

Satis ability problem in description logics with modal operators

Abstract: The paper considers the standard concept description language ALC augmented with various kinds of modal operators which can be applied to concepts and axioms. The main aim is to develop methods of proving decidability of the satis ability problem for this language and apply them to description logics with most important temporal and epistemic operators, thereby obtaining satis ability checking algorithms for these logics. We deal with the possible world semantics under the constant domain assumption and show that the expanding and varying domain assumptions are reducible to it. Models with both nite and arbitrary constant domains are investigated. We begin by considering description logics with only one modal operator and then prove a general transfer theorem which makes it possible to lift the obtained results to many systems of polymodal description logic.

On logics with coimplication

Abstract: This paper investigates (modal) extensions of Heyting–Brouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We first develop matrix as well as Kripke style semantics for those logics. Then, by extending the Godel-embedding of intuitionistic logic into S4 , it is shown that all (modal) extensions of Heyting–Brouwer logic can be embedded into tense logics (with additional modal operators). An extension of the Blok–Esakia-Theorem is proved for this embedding.

Natural Deduction for Non-Classical Logics

Abstract: We present a framework for machine implementation of families of non-classical logics with Kripke-style semantics We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models By appropriate combinations we capture both partial and complete fragments of large families of non-classical logics such as modal, relevance, and intuitionistic logics Our approach is modular and supports uniform proofs of soundness, completeness and proof normalization We have implemented our work in the Isabelle Logical Framework

More Evaluation of Decision Procedures for Modal Logics.

Abstract: This paper follows on previous papers which presented and evaluated various decision procedures for modal logics. It connrms previous experimental results in showing that SAT based decision procedures, i.e., the procedures built on top of decision procedures for propositional satissability, are more ef-cient than tableau based decision procedures. It also connrms previous evidence of an easy-hard-easy pattern in the satissabil-ity curve for modal K. Finally, it provides further experimental results, suggesting that SAT based decision procedures are also more eecient than the decision procedures based on Ohlbach's translation method. Our results contradict some of the results presented in previous papers.

Information Systems, Similarity Relations and Modal Logics

Abstract: In the paper we study two types of information systems: ontological and logical. Systems of ontological type are Property systems and Attribute systems, where the information is represented in terms of the ontological concepts of object, property and attribute. Systems of logical type are Consequence systems and Bi-consequence systems where the information is represented by a collection of sentences, equipped with some inference mechanism. We prove that each Consequence system can be embedded into a certain Property system. The similar results hold for Bi-consequence systems and Attribute systems. These representation theorems are used to give an abstract characterization (by means of a finite set of first-order sentences) of some information relations in Property systems and Attribute systems, including various kinds of similarity relations. Several modal logics with modalities corresponding to some collections of information relations are introduced and their “query meaning” is discussed. One of the main results of the paper are the completeness theorems for the introduced modal logics with respect to their standard semantics.

Encoding modal logics in logical frameworks

Abstract: We present and discuss various formalizations of Modal Logics in Logical Frameworks based on Type Theories. We consider both Hilbert- and Natural Deduction-style proof systems for representing both truth (local) and validity (global) consequence relations for various Modal Logics. We introduce several techniques for encoding the structural peculiarities of necessitation rules, in the typed λ-calculus metalanguage of the Logical Frameworks. These formalizations yield readily proof-editors for Modal Logics when implemented in Proof Development Environments, such as Coq or LEGO.

Optimising Propositional Modal Satisfiability for Description Logic Subsumption

Abstract: Effective optimisation techniques can make a dramatic difference in the performance of knowledge representation systems based on expressive description logics. Because of the correspondence between description logics and propositional modal logic many of these techniques carry over into propositional modal logic satisfiability checking. Currently-implemented representation systems that employ these techniques, such as FaCT and DLP, make effective satisfiable checkers for various propositional modal logics.

Speaking about Transitive Frames in Propositional Languages

Abstract: This paper is a comparative study of the propositional intuitionistic (non-modal) and classical modal languages interpreted in the standard way on transitive frames. It shows that, when talking about these frames rather than conventional quasi-orders, the intuitionistic language displays some unusual features: its expressive power becomes weaker than that of the modal language, the induced consequence relation does not have a deduction theorem and is not protoalgebraic. Nevertheless, the paper develops a manageable model theory for this consequence and its extensions which also reveals some unexpected phenomena. The balance between the intuitionistic and modal languages is restored by adding to the former one more implication.

Comparing Subsumption Optimizations.

Abstract: Effective systems for expressive description logics require a heavily-optimised subsumption checker incorporating a range of optimisation techniques. Because of the correspondence between description logics and propositional modal logic most of these techniques carry over into propositional modal logic satisfiability checking. Some of the techniques are extremely effective on various test suites for propositional modal satisfiability and others are less effective. Further, the effectiveness of a technique depends on the test performed. Description logic systems spend much of their time computing subsumption relationships between descriptions. If the system is based on an expressive description logic then the amount of time spent computing subsumption can be intolerable, even for small knowledge bases, unless steps are taken to heavily optimise this task. The total time spent in subsumption checking comes from the number of subsumption checks required to process a knowledge base as well as from the time spent in performing the hardest of these subsumption checks—with an expressive description logic, the time taken by a small number of hard subsumption checks can dominate

The temporal analysis of Chisholm's paradox

Abstract: Deontic logic, the logic of obligations and permissions, is plagued by several paradoxes that have to be understood before deontic logic can be used as a knowledge representation language. In this paper we extend the temporal analysis of Chisholm's paradox using a deontic logic that combines temporal and preferential notions.

Paraconsistent Modal Logic

Abstract: Generalize the methods of paraconsistent fuzzy reasoning into modal logic, propose a paraconsistent modal logic, whose logical consequence is a modal extension of a paraconsistent fuzzy implication, which has the abilities of both handling inconsistency and representing multi world modals, and present its sound and complete Gentzen style inference system.

The Modal Multilogic of Geometry

Abstract: A spatial logic is a modal logic of which the models are the mathematical models of space. Successively considering the mathematical models of space that are the incidence geometry and the projective geometry, we will successively establish the language, the semantical basis, the axiomatical presentation, the proof of the decidability and the proof of the completeness of INC, the modal multilogic of incidence geometry, and PRO, the modal multilogic of projective geometry.

Labelled Modal Logics: Quantifiers

Abstract: In previous work we gave an approach, based on labelled natural deduction, for formalizing proof systems for a large class of propositional modal logics that includes K, D, T, B, S4, S4.2, KD45, and S5. Here we extend this approach to quantified modal logics, providing formalizations for logics with varying, increasing, decreasing, or constant domains. The result is modular with respect to both properties of the accessibility relation in the Kripke frame and the way domains of individuals change between worlds. Our approach has a modular metatheory too; soundness, completeness and normalization are proved uniformly for every logic in our class. Finally, our work leads to a simple implementation of a modal logic theorem prover in a standard logical framework.

Shakespearian modal logic: a labelled treatment of modal identity

Abstract: In this paper we describe a modal proof system arising from the combination of a tableau-like classical system, which incorporates a restricted ("analytical") version of the cut rule, with a label formalism which allows for a specialised, logic dependant unification algorithm. The system provides a uniform proof-theoretical treatment of first-order (normal) modal logics with identity, with and without Barcan formula and/or its converse.

Modal correspondence for models

Abstract: This paper considers the correspondence theory from modal logic and obtains correspondence results for models as opposed to frames. The key ideas are to consider infinitary modal logic, to phrase correspondence results in terms of substitution instances of a given modal formula, and to identify bisimilar model-world pairs.

A Modular Presentation of Modal Logics in a Logical Framework

Abstract: We present a theoretical and practical approach to the modular natural deduction presentation of modal logics and their implementation in a logical framework Our work treats a large and well known class of modal logics including K KD T B S S S in a uniform way with respect to soundness and completeness for semantics and faithfulness and adequacy of the implementation Moreover it results in a pleasingly simple and usable implementation of these logics x Introduction Logical Frameworks such as the Edinburgh LF and Isabelle have been proposed as a solution to the problem of the explosion of logics and specialized provers for them However it is also acknowledged that this solution is not perfect these frameworks are best suited for encoding well behaved natural deduction formalisms whose metatheory does not deviate too far from the metatheory of the framework logic Modal logics in particular are considered di cult to implement in a clean direct way e g x and Encodings in both the LF and Isabelle have been proposed see section but they have been either Hilbert style or quite specialized and their correctness is subtle We present a method for encoding a large and useful class of propositional modal logics including K KD T B S S S in a natural deduction setting and show once and for all correctness for every encoding in the class We have implemented our work in Isabelle and the result is a simple usable and completely modular natural deduction implementation of these logics Let us consider in more detail the di culty with modal logics since the problem motivates the approach that we pursue The deduction theorem If by adding A as an axiom we can prove B then we can prove A B without A fails in modal logics A semantic explanation of this is that the standard completeness theorem for modal logics says that A i A is true at every world in every suitable Kripke frame hW Ri where W is the set of worlds and R is the accessibility relation Basically A means w W j w A and the deduction theorem states that w W j w A w W j w B w W j w A B where is implication in the meta language and is implication in the object language But this is false we have only w W j w A j w B w W j w A B Thus a naive embedding of a modal logic in a logical framework captures the wrong conse quence relation One solution to this problem is to turn to Hilbert presentations we reject this as it is well known that they are di cult to use in practice Instead motivated by the above semantic account we take the view of a logic as a Labelled Deductive System LDS proposed by Gabbay among others This approach pairs formulae with labels instead of proving A one proves w A where w represents the current world and w W w A i A Then it becomes possible to give a proof theoretic statement of the deduction theorem which is the analogue of the semantic version The same mechanism yields a direct formalization of modal operators like

Modal Deduction in Second-Order Logic and Set Theory - II

Abstract: In this paper, we generalize the set-theoretic translation method for poly-modal logic introduced in [11] to extended modal logics. Instead of devising an ad-hoc translation for each logic, we develop a general framework within which a number of extended modal logics can be dealt with. We first extend the basic set-theoretic translation method to weak monadic second-order logic through a suitable change in the underlying set theory that connects up in interesting ways with constructibility; then, we show how to tailor such a translation to work with specific cases of extended modal logics.

Many-valued and annotated modal logics

Abstract: Many-valued modal logics are of interest from theoretical and practical point of view. Unfortunately, there are no unified theoretical frameworks for many-valued modal logics. We sketch their foundations based on the so-called annotated logics. We give a Kripke semantics for annotated modal logics and prove the completeness theorem. We also discuss possible applications of annotated modal logics to AI.

An Exposition and Development of Kanger’s Early Semantics for Modal Logic

Abstract: Stig Kanger — born of Swedish parents in China in 1924 — was professor of Theoretical Philosophy at Uppsala University from 1968 until his death in 1988. He received his Ph. D. from Stockholm University in 1957 under the supervision of Anders Wedberg. Kanger’s dissertation, Provability in Logic, was remarkably short, only 47 pages, but also very rich in new ideas and results. By combining Gentzen-style techniques with a model theory a la Tarski, Kanger obtained new and simplified proofs of central metalogical results of classical predicate logic: Godel’s completeness theorem, Lowenheim-Skolem’s theorem and Gentzen’s Hauptsatz. The part that had the greatest impact, however, was the 15 pages devoted to modal logic. There Kanger developed a new semantic interpretation for quantified modal logic which had a close family resemblance to semantic theories that were developed around the same time by Jaakko Hintikka, Richard Montague and Saul Kripke (independently of each other and independently of Kanger).

Predicate Logics on Display

Abstract: This chapter provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic. In particular, predicate logics obtained by adopting van Benthem’s modal perspective on first-order logic are considered. The Gentzen systems for these logics augment Belnap’s display logic, DL by introduction rules for the existential and the universal quantifier. These rules for ∀x and ∃x are analogous to the display introduction rules for the modal operators □ and ◊ and do not themselves allow the Barcan formula or its converse to be derived. En route from the minimal ‘modal’ predicate logic to full first-order logic, axiomatic extensions are captured by purely structural sequent rules. The chapter has two main aims, namely 1. presenting a uniform proof-theoretic schema for both substructural subsystems of classical first-order logic, FOL and various subsystems of FOL obtained by relaxing Tarski’s truth definition for the existential and universal quantifiers, and 2. introducing these quantifiers into the framework of DL.

Categorial Inference and Modal Logic

Abstract: This paper establishes a connection between structure sensitive categorial inference and classical modal logic. The embedding theorems for non-associative Lambek Calculus and the whole class of its weak Sahlqvist extensions demonstrate that various resource sensitive regimes can be modelled within the framework of unimodal temporal logic. On the semantic side, this requires decomposition of the ternary accessibility relation to provide its correlation with standard binary Kripke frames and models.