Showing papers on "Normal modal logic published in 1993"

The logic of provability

Abstract: 1. GL and other systems of propositional modal logic 2. Peano arithmetic 3. The box as Bew(x) 4. Semantics for GL and other modal logics 5. Completeness and decidability of GL and K, K4, T, B, S4, and S5 6. Canonical models 7. On GL 8. The fixed point theorem 9. The arithmetical completeness theorems for GL and GLS 10. Trees for GL 11. An incomplete system of modal logic 12. An S4 -preserving proof-theoretical treatment of modality 13. Modal logic within set theory 14. Modal logic within analysis 15. The joint provability logic of consistency and w-consistency 16. On GLB: the fixed point theorem, letterless sentences, and analysis 18. Quantified provability logic with one one-place predicate letter Notes Bibliography Index.

Mathematics of modality

Abstract: Introduction 1. Metamathematics of modal logic 2. Semantic analysis of orthologic 3. Orthomodularity is not elementary 4. Arithmetical necessity, provability and intuitionistic logic 5. Diodorean modality in Minkowski spacetime 6. Grothendieck topology as geometric modality 7. The semantics of Hoare's iteration rule 8. An abstract setting for Henkin proofs 9. A framework for infinitary modal logic 10. The McKinsey axiom is not canonical 11. Elementary logics are canonical and pseudo-equational Bibliography Index.

Extending modal logic

Abstract: modal logic, 121 algebra

Modal nonmonotonic logics: ranges, characterization, computation

Abstract: Many nonmonotonic formalism, including default logic, logic programming with stable models, and autoepistemic logic, can be represented faithfully by means of modal nonmonotonic logics in the family proposed by McDermott and Doyle. In this paper properties of logics in this family are thoroughly investigated. We present several results on characterization of expansions. These results are applicable to a wide class of nonmonotonic modal logics. Using these characterization results, algorithms for computing expansions for finite theories are developed. Perhaps the most important finding of this paper is that the structure of the family of modal nonmonotonic logics is much simpler than that of the family of underlying modal (monotonic) logics. Namely, it is often the case that different monotonic modal logics collapse to the same nonmonotonic system. We exhibit four families of logics whose nonmonotonic variants coincide: 5-KD45, TW5-SW5, N-WK, and W5-D4WB. These nonmonotonic logics naturally represent logics related to commonsense reasoning and knowledge representation such as autoepistemic logic, reflexive autoepistemic logic, default logic, and truth maintenance with negation.

Translation Methods for Non-Classical Logics: An Overview

Abstract: This paper gives an overview on translation methods we have developed for nonclassical logics, in particular for modal logics. Optimized ’functional’ and semi-functional translation into predicate logic is described. Using normal modal logic as an intermediate logic, other logics can be translated into predicate logic as well. As an example, the translation of modal logic of graded modalities is sketched. In the second part of the paper it is shown how to translate Hilbert axioms into properties of the semantic structure and vice versa, i.e. we can automate important parts of correspondence theory. The exact formalisms and the soundness and completeness proofs can be found in the original papers.

Derivation Rules as Anti-Axioms in Modal Logic

Abstract: We discuss a ‘negative’ way of defining frame classes in (multi)modal logic, and address the question of whether these classes can be axiomatized by derivation rules, the ‘non-ξ rules’, styled after Gabbay's Irreflexivity Rule. The main result of this paper is a metatheorem on completeness, of the following kind: If ⋀ is a derivation system having a set of axioms that are special Sahlqvist formulas and ⋀+ is the extension of ⋀ with a set of non-ξ rules, then ⋀+ is strongly sound and complete with respect to the class of frames determined by the axioms and the rules.

Deontic logic as founded on nonmonotonic logic

Abstract: Ever since its inception in the work of von Wright, deontic logic has been developed primarily as a species of modal logic. I argue in this paper, however, that the techniques of nonmonotonic logic may provide a better theoretical framework — at least for the formalization of commensense normative reasoning — than the usual modal treatment. After reviewing some standard approaches to deontic logic, I focus on two areas in which nonmonotonic techniques promise improved understanding: reasoning in the presence of conflicting obligations, and reasoning with conditional obligations.

A modal perspective on the computational complexity of attribute value grammar

Abstract: Many of the formalisms used in Attribute Value grammar are notational variants of languages of propositional modal logic, and testing whether two Attribute Value Structures unify amounts to testing for modal satisfiability. In this paper we put this observation to work. We study the complexity of the satisfiability problem for nine modal languages which mirror different aspects of AVS description formalisms, including the ability to express re-entrancy, the ability to express generalisations, and the ability to express recursive constraints. Two main techniques are used: either Kripke models with desirable properties are constructed, or modalities are used to simulate fragments of Propositional Dynamic Logic. Further possibilities for the application of modal logic in computational linguistics are noted.

Modal logic, transition systems and processes

Abstract: Transition systems can be viewed either as process diagrams or as Kripke structures. The rst perspective is that of process theory, the second that of modal logic. This paper shows how various formalisms of modal logic can be brought to bear on processes. Notions of bisimulation can not only be motivated by operations on transition systems, but they can also be suggested by investigations of modal formalisms. To show that the equational view of processes from process algebra is closely related to modal logic, we consider various ways of looking at the relation between the calculus of basic process algebra and propositional dynamic logic. More concretely, the paper contains preservation results for various bisimulation notions, a result on the expressive power of propositional dynamic logic, and a deenition of bisimulation which is the proper notion of invariance for concurrent propositional dynamic logic.

Cylindric modal logic

Abstract: In this paper we develop a modal formalism called cylindric modal logic we investigate its basic semantics and axiomatics. The motivation for introducing this formalism is twofold: first, it forms an interesting bridge over the gap between propositional formalisms and first-order logic: And second, the modal tools developed in studying cylindric modal logic will be applied to analyze, some problems in algebraic logic.

Modal Logic and Attribute Value Structures

Abstract: This paper examines the relationship between various languages of modal logic and an approach to the specification and processing of natural language grammars currently popular in computational linguistics. This approach is the use of Attribute Value formalisms, and the main aims of the paper are to show that the most common Attribute Value formalisms are nothing but languages of propositional modal logic, and to establish the basic logical theory of the languages concerned.

An Introduction to Executable Modal and Temporal Logics

Abstract: In recent years a number of programming languages based upon the direct execution of either modal or temporal logic formulae have been developed. This use of non-classical logics provides a powerful basis for the representation and implementation of a range of dynamic behaviours. Though many of these languages are still experimental, they are beginning to be applied, not only in Computer Science and AI, but also in less obvious areas such as process control and social modelling.

First-order modal logic theorem proving and functional simulation

Abstract: We propose a translation approach from modal logics to first-order predicate logic which combines advantages from both, the (standard) relational translation and the (rather compact) functional translation method and avoids many of their respective disadvantages (exponential growth versus equality handling). In particular in the application to serial modal logics it allows considerable simplifications such that often even a simple unit clause suffices in order to express the accessibility relation properties. Although we restrict the approach here to first-order modal logic theorem proving it has been shown to be of wider interest, as e.g. sorted logic or terminological logic.

On the definition of modal operators in fuzzy logic

Abstract: The classical modal logic with Kripke semantics is generalized in two ways. First, the set

Undecidability of modal and intermediate first-order logics with two individual variables.

Abstract: The interest in fragments of predicate logics is motivated by the well-known fact that full classical predicate calculus is undecidable (cf. Church [1936]). So it is desirable to find decidable fragments which are in some sense “maximal”, i.e., which become undecidable if they are “slightly” extended. Or, alternatively, we can look for “minimal” undecidable fragments and try to identify the vague boundary between decidability and undecidability. A great deal of work in this area concerning mainly classical logic has been done since the thirties. We will not give a complete review of decidability and undecidability results in classical logic, referring the reader to existing monographs (cf. Suranyi [1959], Lewis [1979], and Dreben, Goldfarb [1979]). A short summary can also be found in the well-known book Church [1956]. Let us recall only several facts. Herein we will consider only logics without functional symbols, constants, and equality.(C1) The fragment of the classical logic with only monadic predicate letters is decidable (cf. Behmann [1922]).(C2) The fragment of the classical logic with a single binary predicate letter is undecidable. (This is a consequence of Godel [1933].)(C3) The fragment of the classical logic with a single individual variable is decidable; in fact it is equivalent to Lewis S5 (cf. Wajsberg [1933]).(C4) The fragment of the classical logic with two individual variables is decidable (Segerberg [1973] contains a proof using modal logic; Scott [1962] and Mortimer [1975] give traditional proofs.)(C5) The fragment of the classical logic with three individual variables and binary predicate letters is undecidable (cf. Suranyi [1943]). In fact this paper considers formulas of the following typeφ,ψ being quantifier-free and the set of binary predicate letters which can appear in φ or ψ being fixed and finite.

Generalized quantifiers and modal logic

Abstract: We study several modal languages in which some (sets of) generalized quantifiers can be represented; the main language we consider is suitable for defining any first order definable quantifier, but we also consider a sublanguage thereof, as well as a language for dealing with the modal counterparts of some higher order quantifiers. These languages are studied both from a modal logic perspective and from a quantifier perspective. Thus the issues addressed include normal forms, expressive power, completeness both of modal systems and of systems in the quantifier tradition, complexity as well as syntactic characterizations of special semantic constraints. Throughout the paper several techniques current in the theory of generalized quantifiers are used to obtain results in modal logic, and conversely.

Measure-Based Semantics for Modal Logic

Abstract: A fuzzy-measure-based approach to semantics for modal logic is presented and its several properties are discussed. Measure-based models for modal logic are defined and the soundness and completeness theorems of several systems of modal logic are proved with respect to classes of finite measure-based models, particularly, formulated by fuzzy, possibility, necessity, probability, and Dirac measures.

Optimized Translation of Multi Modal Logic into Predicate Logic

Abstract: The functional translation from modal logic into first-order predicate logic is revised. Quantifier elimination of second-order predicates is used to translate almost arbitrary modal systems, i.e. not only modal formulae, but also characteristic Hilbert axioms, fully automatically into predicate logic. Various optimizations of the functional translation are investigated. They even permit the translation of modal systems like the McKinsey axiom, whose correspondence property of the accessibility relation is not first-order axiomatizable. Quantifier exchange rules are defined for modifying the translated Hilbert axioms and thus simplifying their transformation into theory unification algorithms.

La connaissance commune en logique modale

Abstract: The problem of Common Knowledge will be considered in two classes of models: a class K* of Kripke models and a class S of Scott models. Two modal logic systems will be defined. Those systems, KC and MC, include an axiomatisation of Common Knowledge. We prove determination of each system by the corresponding class of models.

Formal systems for modal operators on locales

Abstract: In the paper [8], the first author developped a topos- theoretic approach to reference and modality. (See also [5]). This approach leads naturally to modal operators on locales (or ‘spaces without points”). The aim of this paper is to develop the theory of such modal operators in the context of the theory of locales, to axiomatize the propositional modal logics arising in this context and to study completeness and decidability of the resulting systems.

The semantics ofR4

Abstract: The LogicR4 is obtained by adding the axiom ▭(A vB→(◊Av▭B) to the modal relevant logicNR. We produce a model theory for this logic and show completeness. We also show that there is a natural embedding of a Kripke model forS4 in eachR4 model structure.

A modal logic of quantification and substitution

Abstract: The aim of this paper is to study the n-variable fragment of first order logic from a modal perspective. We define a modal formalism called cylindric mirror modal logic, and show how it is a modal version of first order logic with substitution. In this approach, we can define a semantics for the language which is closely related to algebraic logic, as we find Polyadic Equality Algebras as the modal or complex algebras of our system. The main contribution of the paper is a characterization of the intended 'mirror cubic' frames of the formalisms and, a consequence of the special form of this characterization, a completeness theorem for these intended frames. As a consequence, we find complete finite yet unorthodox derivation systems for the equational theory of finite-dimensional representable polyadic equality algebras.

A Comparative Fuzzy Modal Logic

Abstract: A formal logical systems dealing both with uncertainty (possibility) and vagueness (fuzziness) is investigated. It is many-valued and modal. The system is related to a many-valued tense logic. A completeness theorem is exhibited.

Prefinitely Axiomatizable Modal and Intermediate Logics

Abstract: A logic Λ bounds a property P if all proper extensions of Λ have P while Λ itself does not. We construct logics bounding finite axiomatizability and logics bounding finite model property in the lattice of intermediate logics and the lattice of normal extensions of K4.3.

Classically complete modal relevant logics

Abstract: A variety of modal logics based on the relevant logic R are presented. Models are given for each of these logics and completeness is shown. It is also shown that each of these logics admits Ackermann's rule γ and as a corollary of this it is proved that each logic is a conservative extension of its counterpart based on classical logic, hence we call them “classically complete”. MSC: 03B45, 03B46.