Showing papers on "Normal modal logic published in 2005"

Proof Analysis in Modal Logic

Abstract: A general method for generating contraction- and cut-free sequent calculi for a large family of normal modal logics is presented. The method covers all modal logics characterized by Kripke frames determined by universal or geometric properties and it can be extended to treat also Godel–Lob provability logic. The calculi provide direct decision methods through terminating proof search. Syntactic proofs of modal undefinability results are obtained in the form of conservativity theorems.

Model theory for extended modal languages

Abstract: 207

Expressivity of coalgebraic modal logic: the limits and beyond

Abstract: Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modeled as coalgebras. Logics with modal operators obtained from so-called predicate liftings have been shown to be invariant under behavioral equivalence. Expressivity results stating that, conversely, logically indistinguishable states are behaviorally equivalent depend on the existence of separating sets of predicate liftings for the signature functor at hand. Here, we provide a classification result for predicate liftings which leads to an easy criterion for the existence of such separating sets, and we give simple examples of functors that fail to admit expressive normal or monotone modal logics, respectively, or in fact an expressive (unary) modal logic at all. We then move on to polyadic modal logic, where modal operators may take more than one argument formula. We show that every accessible functor admits an expressive polyadic modal logic. Moreover, expressive polyadic modal logics are, unlike unary modal logics, compositional.

Interpolation and Definability: Modal and Intuitionistic Logic

Abstract: 1. Introduction and Discussion 2. Modal and Superintuitionistic Logics: Basic Concepts 3. Superintuitionistic Logics and Normal Extensions of the Modal Logics S4 4. The Interpolation Theorem in Intuitionistic Predicate Calculus 5. Interpolation and Definability in Quantified Logics 6. Craig's Theorem in Superintuitionistic Logics and Amalgamable Varieties of Pseudoboolean Algebras 7. Interpolation, Definability, Amalgamation 8. Interpolation in Normal Extensions of the Modal Logic S4 9. Complexity of Some Problems in Modal and Intuitionistic Calculi 10. Interpolation in Modal Infinite Slice Logics Containing the Logic K4 11. An Analog of Beth's Theorem in Normal Extensions of the Modal Logic K4 12. Extensions of the Provability Logic 13. Syntactic Proof of Interpolation for the Intuitionistic Predicate Logic 14. Interpolation by Translation 15. Interpolation in (Intuitionistic) Logic Programming 16. Interpolation in Goal-directed Proof Systems 17. Further Results and Discussion Appendix References Index

The Undecidability of Iterated Modal Relativization

Abstract: In dynamic epistemic logic and other fields, it is natural to consider relativization as an operator taking sentences to sentences. When using the ideas and methods of dynamic logic, one would like to iterate operators. This leads to iterated relativization. We are also concerned with the transitive closure operation, due to its connection to common knowledge. We show that for three fragments of the logic of iterated relativization and transitive closure, the satisfiability problems are fi1 Σ11–complete. Two of these fragments do not include transitive closure. We also show that the question of whether a sentence in these fragments has a finite (tree) model is fi0 Σ01–complete. These results go via reduction to problems concerning domino systems.

Admissible Rules of Modal Logics

Abstract: We construct explicit bases of admissible rules for a representative class of normal modal logics (including the systems K4, GL, S4, Grz, and GL.3), by extending the methods of S. Ghilardi and R. Iemhoff. We also investigate the notion of admissible multiple conclusion rules.

Bounded distributive lattices with strict implication

Abstract: The present paper introduces and studies the variety WH of weakly Heyting algebras. It corresponds to the strict implication fragment of the normal modal logic K which is also known as the subintuitionistic local consequence of the class of all Kripke models. The tools developed in the paper can be applied to the study of the subvarieties of WH; among them are the varieties determined by the strict implication fragments of normal modal logics as well as varieties that do not arise in this way as the variety of Basic algebras or the variety of Heyting algebras. Apart from WH itself the paper studies the subvarieties of WH that naturally correspond to subintuitionistic logics, namely the variety of R-weakly Heyting algebras, the variety of T-weakly Heyting algebras and the varieties of Basic algebras and subresiduated lattices. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Nearly every normal modal logic is paranormal

Abstract: An overcomplete logic is a logic that ‘ceases to make the difference’: According to such a logic, all inferences hold independently of the nature of the statements involved. A negation-inconsistent logic is a logic having at least one model that satisfies both some statement and its negation. A negation-incomplete logic has at least one model according to which neither some statement nor its negation are satisfied. Paraconsistent logics are negation-inconsistent yet non-overcomplete; paracomplete logics are negation-incomplete yet non-overcomplete. A paranormal logic is simply a logic that is both paraconsistent and paracomplete. Despite being perfectly consistent and complete with respect to classical negation, nearly every normal modal logic, in its ordinary language and interpretation, admits to some latent paranormality: It is paracomplete with respect to a negation defined as an impossibility operator, and paraconsistent with respect to a negation defined as non-necessity. In fact, as it will be shown here, even in languages without a primitive classical negation, normal modal logics can often be alternatively characterized directly by way of their paranormal negations and related operators. So, instead of talking about ‘necessity’, ‘possibility’, and so on, modal logics could be seen just as devices tailored for the study of (modal) negation. This paper shows how and to what extent this alternative characterization of modal logics can be realized.

Some Results on Modal Axiomatization and Definability for Topological Spaces

Abstract: We consider two topological interpretations of the modal diamond—as the closure operator (C-semantics) and as the derived set operator (d-semantics). We call the logics arising from these interpretations C-logics and d-logics, respectively. We axiomatize a number of subclasses of the class of nodec spaces with respect to both semantics, and characterize exactly which of these classes are modally definable. It is demonstrated that the d-semantics is more expressive than the C-semantics. In particular, we show that the d-logics of the six classes of spaces considered in the paper are pairwise distinct, while the C-logics of some of them coincide.

Negation in the Context of Gaggle Theory

Abstract: We study an application of gaggle theory to unary negative modal operators. First we treat negation as impossibility and get a minimal logic system Ki that has a perp semantics. Dunn's kite of different negations can be dealt with in the extensions of this basic logic Ki. Next we treat negation as “unnecessity” and use a characteristic semantics for different negations in a kite which is dual to Dunn's original one. Ku is the minimal logic that has a characteristic semantics. We also show that Shramko's falsification logic FL can be incorporated into some extension of this basic logic Ku. Finally, we unite the two basic logics Ki and Ku together to get a negative modal logic K-, which is dual to the positive modal logic K+ in [7]. Shramko has suggested an extension of Dunn's kite and also a dual version in [12]. He also suggested combining them into a “united” kite. We give a united semantics for this united kite of negations.

On the complexity of hybrid logics with binders

Abstract: Hybrid logic refers to a group of logics lying between modal and first-order logic in which one can refer to individual states of the Kripke structure. In particular, the hybrid logic HL(@, ↓ ) is an appealing extension of modal logic that allows one to refer to a state by means of the given names and to dynamically create new names for a state. Unfortunately, as for the richer first-order logic, satisfiability for the hybrid logic HL(@, ↓ ) is undecidable and model checking for HL(@, ↓ ) is PSpace-complete. We carefully analyze these results and we isolate large fragments of HL(@, ↓ ) for which satisfiability is decidable and model checking is below PSpace.

Interpolation for extended modal languages

Abstract: Several extensions of the basic modal language are characterized in terms of interpolation. Our main results are of the following form: Language ℒ' is the least expressive extension of ℒ with interpolation. For instance, let ℳ(D) be the extension of the basic modal language with a difference operator [7]. First-order logic is the least expressive extension of ℳ(D) with interpolation. These characterizations are subsequently used to derive new results about hybrid logic, relation algebra and the guarded fragment.

Ultrafilter extensions for coalgebras

Abstract: This paper studies finitary modal logics as specification languages for Set-coalgebras (coalgebras on the category of sets) using Stone duality. It is well-known that Set-coalgebras are not semantically adequate for finitary modal logics in the sense that bisimilarity does not in general coincide with logical equivalence. Stone-coalgebras (coalgebras over the category of Stone spaces), on the other hand, do provide an adequate semantics for finitary modal logics. This leads us to study the relationship of finitary modal logics and Set-coalgebras by uncovering the relationship between Set-coalgebras and Stone-coalgebras. This builds on a long tradition in modal logic, where one studies canonical extensions of modal algebras and ultrafilter extensions of Kripke frames to account for finitary logics. Our main contributions are the generalisations of two classical theorems in modal logic to coalgebras, namely the Jonsson-Tarski theorem giving a set-theoretic representation for each modal algebra and the bisimulation-somewhere-else theorem stating that two states of a coalgebra have the same (finitary modal) theory iff they are bisimilar (or behaviourally equivalent) in the ultrafilter extension of the coalgebra.

On epistemic logic with justification

Abstract: The true belief components of Plato's tripartite definition of knowledge as justified true belief are represented in formal epistemology by modal logic and its possible worlds semantics. At the same time, the justification component of Plato's definition did not have a formal representation. This paper introduces the notion of justification into formal epistemology. Epistemic logic with justification, along with the usual knowledge operator sF (F is known), contains assertions t:F (t is a justification for F). We suggest an epistemic semantics which augments Kripke models with a natural Fitting-style treatment of justification assertions t:F. Completeness and some new specific properties of basic systems of epistemic logic with justification are established.

Fuzzy Modal Logics

Abstract: In the paper we introduce formal calculi which are a generalization of propositional modal logics. These calculi are called fuzzy modal logics. We introduce the concept of a fuzzy Kripke model and consider a semantics of these calculi in the class of fuzzy Kripke models. The main result of the paper is the completeness theorem of a minimal fuzzy modal logic in the class of fuzzy Kripke models.

A decision procedure for the alternation-free two-way modal µ-calculus

Abstract: The satisfiability checking problem is known to be decidable for a variety of modal/temporal logics such as the modal μ-calculus, but effective implementation has not necessarily been developed for all such logics. In this paper, we propose a decision procedure using the tableau method for the alternation-free two-way modal μ-calculus. Although the size of the tableau set maintained in the method might be large for complex formulas, the set and the operations on it can be expressed using BDD and therefore we can implement the method in an effective way.

A Modal Analysis of Presupposition and Modal Subordination

Abstract: In this paper I will give a modal two-dimensional analysis of presupposition and modal subordination. I will think of presupposition as a non-veridical propositional attitude. This allows me to evaluate what is presupposed and what is asserted at different dimensions without getting into the binding problem. What is presupposed will be represented by an accessibility relation between possible worlds. The major part of the paper consists of a proposal to account for the dependence of the interpretation of modal expressions, i.e. modal subordination, in terms of an accessibility relation as well. Moreover, I show how such an analysis can be extended from the propositional to the predicate logical level.

Minimal predicates, fixed-points, and definability

Abstract: Minimal predicates P satisfying a given first-order description ϕ ( P ) occur widely in mathematical logic and computer science. We give an explicit first-order syntax for special first-order ‘ PIA conditions’ ϕ ( P ) which guarantees unique existence of such minimal predicates. Our main technical result is a preservation theorem showing PIA -conditions to be expressively complete for all those first-order formulas that are preserved under a natural model-theoretic operation of ‘predicate intersection’. Next, we show how iterated predicate minimization on PIA -conditions yields a language MIN( FO ) equal in expressive power to LFP( FO ), first-order logic closed under smallest fixed-points for monotone operations. As a concrete illustration of these notions, we show how our sort of predicate minimization extends the usual frame correspondence theory of modal logic, leading to a proper hierarchy of modal axioms: first-order-definable, first-order fixed-point definable, and beyond.

Undecidability of first-order intuitionistic and modal logics with two variables

Abstract: We prove that the two-variable fragment of first-order intuitionistic logic is undecidable, even without constants and equality. We also show that the two-variable fragment of a quantified modal logic L with expanding first-order domains is undecidable whenever there is a Kripke frame for L with a point having infinitely many successors (such are, in particular, the first-order extensions of practically all standard modal logics like K, K4, GL, S4, S5, K4.1, S4.2, GL.3, etc.). For many quantified modal logics, including those in the standard nomenclature above, even the monadic two-variable fragments turn out to be undecidable.

A modal logic framework for multi-agent belief fusion

Abstract: This article provides a modal logic framework for reasoning about multi-agent belief and its fusion. We propose logics for reasoning about cautiously merged agent beliefs that have different degrees of reliability. These logics are obtained by combining the multi-agent epistemic logic and multi-source reasoning systems. The fusion is cautious in the sense that if an agent's belief is in conflict with those of higher priorities, then his belief is completely discarded from the merged result. We consider two strategies for the cautious merging of beliefs. In the first, called level cutting fusion, if inconsistency occurs at some level, then all beliefs at the lower levels are discarded simultaneously. In the second, called level skipping fusion, only the level at which the inconsistency occurs is skipped. We present the formal semantics and axiomatic systems for these two strategies and discuss some applications of the proposed logical systems. We also develop a tableau proof system for the logics and prove the complexity result for the satisfiability and validity problems of these logics.

Elementary Canonical Formulae: A Survey on Syntactic, Algorithmic, and Model-theoretic Aspects

Abstract: In terms of validity in Kripke frames, a modal formula expresses a universal monadic second-order condition. Those modal formulae which are equivalent to first-order conditions are called ele- mentary. Modal formulae which have a certain persistence property which implies their validity in all canonical frames of modal logics axiomatized with them, and therefore their completeness, are called canonical. This is a survey of a recent and ongoing study of the class of elementary and canonical modal formulae. We summarize main ideas and results, and outline further research perspectives.

Logics of essence and accident

Abstract: We say that things happen accidentally when they do indeed happen, but only by chance. In the opposite situation, an essential happening is inescapable, its inevitability being the sine qua non for its very occurrence. This paper will investigate modal logics on a language tailored to talk about essential and accidental statements. Completeness of some among the weakest and the strongest such systems is attained. The weak expressibility of the classical propositional language enriched with the non-normal modal operators of essence and accident is highlighted and illustrated, both with respect to the definability of the more usual modal operators as well as with respect to the characterizability of classes of frames. Several interesting problems and directions are left open for exploration.

Modal logic investigations in the semantics of counts-as

Abstract: The work investigates the logic underlying the representation of the non-regulative component of normative systems, the so-called counts-as The analytic thesis we hold here is to view counts-as statements as statements which yield classifications and which hold only with respect to a context These two aspects of the semantics of counts-as-the classificatory flavor, and the contextual character-are then investigated by means of modal logic techniques from a semantics-driven perspective, and a formalization of counts-as statements is thus proposed The result is then compared in detail with previous work on the topic, and related with work which, despite developed in different areas of applied and philosophical logic, shares interesting technical and theoretical similarities with our proposal

Ground Nonmonotonic Modal Logic S5: New Results

Abstract: We study logic programs under Gelfond's translation in the context of modal logic S5. We show that for arbitrary logic programs (propositional theories where logic negation is associated with default negation) ground nonmonotonic modal logics between T and S5 are equivalent. Furthermore, we also show that these logics are equivalent to a nonmonotonic logic that we construct using the well known F O U R bilattice. We will call this semantic GNM-S5 as a reminder of its origin in the logic S5. Finally we show that, for normal programs, our approach is closely related to theWell-Founded-by-Cases Semantics introduced by Schlipf and the WFS+ proposed by Dix. We prove that GNM-S5 has the properties of classicality and extended cut. While WFS+ also supports classicality it fails to satisfy the extended cut principle, an important property available in other semantics such as stable models. Hence, we claim that GNM-S5 is a good candidate for defining a nonmonotonic semantics closer to the direction of classical logic.

Coalgebraic modal logic of finite rank

Abstract: This paper studies coalgebras from the perspective of finite observations. We introduce the notion of finite step equivalence and a corresponding category with finite step equivalence-preserving morphisms. This category always has a final object, which generalises the canonical model construction from Kripke models to coalgebras. We then turn to logics whose formulae are invariant under finite step equivalence, which we call logics of rank $\omega$. For these logics, we use topological methods and give a characterisation of compact logics and definable classes of models.

The least fibred lifting and the expressivity of coalgebraic modal logic

Abstract: Every endofunctor B on the category Set can be lifted to a fibred functor on the category (fibred over Set) of equivalence relations and relation-preserving functions. In this paper, the least (fibre-wise) of such liftings, L(B), is characterized for essentially any B. The lifting has all the useful properties of the relation lifting due to Jacobs, without the usual assumption of weak pullback preservation; if B preserves weak pullbacks, the two liftings coincide. Equivalence relations can be viewed as Boolean algebras of subsets (predicates, tests). This correspondence relates L(B) to the least test suite lifting T(B), which is defined in the spirit of predicate lifting as used in coalgebraic modal logic. Properties of T(B) translate to a general expressivity result for a modal logic for B-coalgebras. In the resulting logic, modal operators of any arity can appear.