Showing papers on "Normal modal logic published in 2006"

Modal Logic for Philosophers

Abstract: This book on modal logic is especially designed for philosophy students. It provides an accessible yet technically sound treatment of modal logic and its philosophical applications. Every effort is made to simplify the presentation by using diagrams instead of more complex mathematical apparatus. These and other innovations provide philosophers with easy access to a rich variety of topics in modal logic, including a full coverage of quantified modal logic, non-rigid designators, definite descriptions, and the de-re de-dicto distinction. Discussion of philosophical issues concerning the development of modal logic is woven into the text. The book uses natural deduction systems, which are widely regarded as the easiest to teach and use. It also includes a diagram technique that extends the method of truth trees to modal logic. This provides a foundation for a novel method for showing completeness that is easy to extend to quantifiers. This second edition contains a new chapter on logics of conditionals, an updated and expanded bibliography, and is updated throughout.

Pure extensions, proof rules, and hybrid axiomatics

Abstract: In this paper we argue that hybrid logic is the deductive setting most natural for Kripke semantics. We do so by investigating hybrid axiomatics for a variety of systems, ranging from the basic hybrid language (a decidable system with the same complexity as orthodox propositional modal logic) to the strong Priorean language (which offers full first-order expressivity). We show that hybrid logic offers a genuinely first-order perspective on Kripke semantics: it is possible to define base logics which extend automatically to a wide variety of frame classes and to prove completeness using the Henkin method. In the weaker languages, this requires the use of non-orthodox rules. We discuss these rules in detail and prove non-eliminability and eliminability results. We also show how another type of rule, which reflects the structure of the strong Priorean language, can be employed to give an even wider coverage of frame classes. We show that this deductive apparatus gets progressively simpler as we work our way up the expressivity hierarchy, and conclude the paper by showing that the approach transfers to first-order hybrid logic.

Lattices of intermediate and cylindric modal logics

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Logics with Common Weak Completions

Abstract: We introduce the notion of X-stable models parametrized by a given logic X. Such notion is based on a construction that we call weak completions: a set of atoms M is an X-stable model of a theory T if M is a model of T, in the sense of classical logic, and the weak completion of T (namely T[:e M) can prove, in the sense given by logic X, every atom in the set M. We prove that, for normal logic programs, the result obtained by these weak completions is invariant with respect to a large family of logics. Two kinds of logics are mainly considered: paraconsistent logics and normal modal logics. For modal logics we use a translation proposed by Gelfond that identifies:œa with:a. As a consequence we prove that several semantics (recently introduced) for non-monotonic reasoning (NMR) are equivalent for normal programs. In addition, we show that such semantics can be characterized by a fixed-point operator. Also, as a side effect, we provide new results for the stable model semantics.

The logic of being informed

Abstract: One of the open problems in the philosophy of information is whether there is an information logic (IL), different from epistemic (EL) and doxastic logic (DL), which formalises the relation ia is informed that pi (Iap) satisfactorily. In this paper, the problem is solved by arguing that the axiom schemata of the normal modal logic (NML) KTB (also known as B or Br or Brouwer’s system) are well suited to formalise the relation of ibeing informedi. After having shown that IL can be constructed as an informational reading of KTB, four consequences of a KTB-based IL are explored: information overload; the veridicality thesis (Iap ! p); the relation between IL and EL; and theKp ! Bp principle or entailment property, according to which knowledge implies belief. Although these issues are discussed later in the article, they are the motivations behind the development of IL.

Modal Frame Correspondences and Fixed-Points

Abstract: Taking Lob's Axiom in modal provability logic as a running thread, we discuss some general methods for extending modal frame correspondences, mainly by adding fixed-point operators to modal languages as well as their correspondence languages. Our suggestions are backed up by some new results – while we also refer to relevant work by earlier authors. But our main aim is advertizing the perspective, showing how modal languages with fixed-point operators are a natural medium to work with.

Bisimulation quantifiers for modal logics

Abstract: Modal logics have found applications in many different contexts. For example, epistemic modal logics can be used to reason about security protocols, temporal modal logics can be used to reason about the correctness of distributed systems and propositional dynamic logic can reason about the correctness of programs. However, pure modal logic is expressively weak and cannot represent many interesting secondorder properties that are expressible, for example, in the μ-calculus.

Evidence reconstruction of epistemic modal logic S5

Abstract: We introduce the logic of proofs whose modal counterpart is the modal logic S5. The language of Logic of Proofs LP is extended by a new unary operation of negative checker “?”. We define Kripke-style models for the resulting logic in the style of Fitting models and prove the corresponding Completeness theorem. The main result is the Realization theorem for the modal logic S5.

The modalized Heyting calculus: a conservative modal extension of the Intuitionistic Logic ★

Abstract: In this paper we define an augmentation mHC of the Heyting propositional calculus HC by a modal operator □. This modalized Heyting calculus mHC is a weakening of the Proof-Intuitionistic Logic KM of Kuznetsov and Muravitsky. In Section 2 we present a short selection of attractive (algebraic, relational, topological and categorical) features of mHC. In Section 3 we establish some close connections between mHC and certain normal extension K4.Grz of the modal system K4. We define a translation of mHC into K4.Grz and prove that this translation is exact, i. e. theorem-preserving and deducibility-invariant. We have established (however, in this note we do not present a proof of this) that the lattice of all extensions of mHC is isomorphic to the lattice of normal extensions of K4.Grz (a generalization of the Kuznetsov and Muravitsky theorem).

First-order classical modal logic

Abstract: The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40]. We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics.

A new combination procedure for the word problem that generalizes fusion decidability results in modal logics

Abstract: Previous results for combining decision procedures for the word problem in the non-disjoint case do not apply to equational theories induced by modal logics-which are not disjoint for sharing the theory of Boolean algebras. Conversely, decidability results for the fusion of modal logics are strongly tailored towards the special theories at hand, and thus do not generalize to other types of equational theories. In this paper, we present a new approach for combining decision procedures for the word problem in the non-disjoint case that applies to equational theories induced by modal logics, but is not restricted to them. The known fusion decidability results for modal logics are instances of our approach. However, even for equational theories induced by modal logics our results are more general since they are not restricted to so-called normal modal logics.

Counts-as: classification or constitution? an answer using modal logic

Abstract: By making use of modal logic techniques, the paper disentangles two semantically different readings of statements of the type X counts as Y in context C (the classificatory and the constitutive readings) showing that, in fact, ‘counts-as is said in many ways'.

The Dynamics of Syntactic Knowledge

Abstract: The syntactic approach to epistemic logic avoids the logical omniscience problem by taking knowledge as primary rather than as defined in terms of possible worlds. In this study, we combine the syntactic approach with modal logic, using transition systems to model reasoning. We use two syntactic epistemic modalities: ‘knowing at least’ a set of formulae and ‘knowing at most’ a set of formulae. We are particularly interested in models restricting the set of formulae known by an agent at a point in time to be finite. The resulting systems are investigated from the point of view of axiomatization and complexity. We show how these logics can be used to formalise non-omniscient agents who know some inference rules, and study their relationship to other systems of syntactic epistemic logics,

Generalized modal satisfiability

Abstract: It is well-known that modal satisfiability is PSPACE-complete [Lad77]. However, the complexity may decrease if we restrict the set of propositional operators used. Note that there exist an infinite number of propositional operators, since a propositional operator is simply a Boolean function. We completely classify the complexity of modal satisfiability for every finite set of propositional operators, i.e., in contrast to previous work, we classify an infinite number of problems. We show that, depending on the set of propositional operators, modal satisfiability is PSPACE-complete, coNP-complete, or in P. We obtain this trichotomy not only for modal formulas, but also for their more succinct representation using modal circuits.

Completeness results for many-valued \Lukasiewicz modal systems and relational semantics

Abstract: The paper is dedicated to the problem of adding a modality to the \Lukasiewicz many-valued logics in the purpose of obtaining completeness results for Kripke semantics. We define a class of modal many-valued logics and their corresponding Kripke models and modal many-valued algebras. Completeness results are considered through the construction of a canonical model. Completeness is obtained for modal finitely-valued logics but also for a modal many-valued system with an infinitary deduction rule. We introduce two classes of frames for the finitely-valued logics and show that they define two distinct classes of Kripke-complete logics.

Expressivity of Second Order Propositional Modal Logic

Abstract: We consider second-order propositional modal logic (SOPML), an extension of the basic modal language with propositional quantifiers introduced by Kit Fine in 1970. We determine the precise expressive power of SOPML by giving analogues of the Van Benthem–Rosen theorem and the Goldblatt Thomason theorem. Furthermore, we show that the basic modal language is the bisimulation invariant fragment of SOPML, and we characterize the bounded fragment of first-order logic as being the intersection of first-order logic and SOPML.

PSPACE Bounds for Rank-1 Modal Logics

Abstract: For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank-1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatization, in PSPACE. This leads not only to a unified derivation of (known) tight PSPACE-bounds for a number of logics including K, coalition logic, and graded modal logic (and to a new algorithm in the latter case), but also to a previously unknown tight PSPACE-bound for probabilistic modal logic, with rational probabilities coded in binary. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.

A Reconstruction of Aristotle's Modal Syllogistic

Abstract: Ever since Łukasiewicz, it has been opinio communis that Aristotle's modal syllogistic is incomprehensible due to its many faults and inconsistencies, and that there is no hope of finding a single consistent formal model for it. The aim of this paper is to disprove these claims by giving such a model. My main points shall be, first, that Aristotle's syllogistic is a pure term logic that does not recognize an extra syntactic category of individual symbols besides syllogistic terms and, second, that Aristotelian modalities are to be understood as certain relations between terms as described in the theory of the predicables developed in the Topics. Semantics for modal syllogistic is to be based on Aristotelian genus-species trees. The reason that attempts at consistently reconstructing modal syllogistic have failed up to now lies not in the modal syllogistic itself, but in the inappropriate application of modern modal logic and extensional set theory to the modal syllogistic. After formalizing the underlying...

A finite model construction for coalgebraic modal logic

Abstract: In recent years, a tight connection has emerged between modal logic on the one hand and coalgebras, understood as generic transition systems, on the other hand. Here, we prove that (finitary) coalgebraic modal logic has the finite model property. This fact not only reproves known completeness results for coalgebraic modal logic, which we push further by establishing that every coalgebraic modal logic admits a complete axiomatization of rank 1; it also enables us to establish a generic decidability result and a first complexity bound. Examples covered by these general results include, besides standard Hennessy-Milner logic, graded modal logic and probabilistic modal logic.

Presburger modal logic is PSPACE-Complete

Abstract: We introduce a Presburger modal logic PML with regularity constraints and full Presburger constraints on the number of children that generalize graded modalities, also known as number restrictions in description logics We show that PML satisfiability is only pspace-complete by designing a Ladner-like algorithm that can be turned into an analytic proof system algorithm This extends a well-known and non-trivial pspace upper bound for graded modal logic Furthermore, we provide a detailed comparison with logics that contain Presburger constraints and that are dedicated to query XML documents As an application, we show that satisfiability for Sheaves Logic SL is pspace-complete, improving significantly its best known upper bound

The truth about algorithmic problems in correspondence theory

Abstract: We present proofs that the following three major algorith- mic problems in Correspondence Theory are undecidable: first-order defin- ability of propositional modal formulas, modal (propositional) definability of first-order (classical predicate) formulas, and the correspondence prob- lem (equivalence on Kripke frames) for propositional modal formulas and first-order formulas. These results have been known since Chagrova's PhD thesis (4) published in 1989. However, (4) considered (propositional) intu- itionistic formulas which are much harder to deal with than the modal ones. Although the proofs from (4) provided some extra subtle results (such as the undecidability of the set of propositional formulas that are first-order definable on the class of countable frames, but not on the class of all frames; see also (9)), the technique employed was rather involved and the intuition behind was not clear (which has been mentioned by many of our colleagues). The main aim of this paper is to give simple and direct proofs for the modal case and to consider some variations of the three major problems (e.g., the case of countable frames). Our second goal is to show that in the proof of the undecidability of modal definability of first-order formulas, it is sufficient to use one simple modal formula, namely?.

Algorithmic Correspondence and Completeness in Modal Logic. II. Polyadic and Hybrid Extensions of the Algorithm SQEMA

Abstract: In Conradie, Goranko, and Vakarelov (2006, Logical Methods in Computer Science, 2) we introduced a new algorithm, , for computing first-order equivalents and proving the canonicity of modal formulae of the basic modal language. Here we extend , first to arbitrary and reversive polyadic modal languages, and then to hybrid polyadic languages too. We present the algorithm, illustrate it with some examples, and prove its correctness with respect to local equivalence of the input and output formulae, its completeness with respect to the polyadic inductive formulae introduced in Goranko and Vakarelov (2001, J. Logic. Comput., 11, 737–754) and Goranko and Vakarelov (2006, Ann. Pure. Appl. Logic, 141, 180–217), and the d-persistence (with respect to descriptive frames) of the formulae on which the algorithm succeeds. These results readily expand to completeness with respect to hybrid inductive polyadic formulae and di-persistence (with respect to discrete frames) in hybrid reversive polyadic languages.

Conservative extensions in modal logic

Abstract: Every normal modal logic L gives rise to the consequence relation ' |=L which holds if, and only if, is true in a world of an L-model whenever ' is true in that world. We consider the following al- gorithmic problem for L. Given two modal formulas '1 and '2, decide whether '1^'2 is a conservative extension of'1 in the sense that whenever '1 ^'2 |=L and does not contain propositional variables not occurring in '1, then '1 |=L. We first prove that the conservativeness problem is coNExpTime-hard for all modal logics of unbounded width (which have rooted frames with more than N successors of the root, for any N < !). Then we show that this problem is (i) coNExpTime-complete for S5 and K, (ii) in ExpSpace for S4 and (iii) ExpSpace-complete for GL.3 (the logic of finite strict linear orders). The proofs for S5 and K use the fact that these logics have uniform interpolants of exponential size.

Presburger modal logic is PSPACE-complete

Abstract: We introduce a Presburger modal logic PML with regularity constraints and full Presburger constraints on the number of children that generalize graded modalities, also known as number restrictions in description logics. We show that PML satisfiability is only PSPACE-complete by designing a Ladner-like algorithm. This extends a well-known and non-trivial PSPACE upper bound for graded modal logic. Furthermore, we provide a detailed comparison with logics that contain Presburger constraints and that are dedicated to query XML documents. As an application, we show that satisfiability for Sheaves Logic SL is PSPACE-complete, improving significantly its best known upper bound.

Semisimple varieties of modal algebras

Abstract: In this paper we show that a variety of modal algebras of finite type is semisimple iff it is discriminator iff it is both weakly transitive and cyclic. This fact has been claimed already in [4] (based on joint work by the two authors) but the proof was fatally flawed.

A Hybrid Intuitionistic Logic: Semantics and Decidability

Abstract: We study a hybrid intuitionistic modal logic suitable for reasoning about distribution of resources. The modalities of the logic allow validation of properties in a particular place, in some place and in all places. We provide a sound and complete Kripke semantics. We also define a sound and complete birelational semantics, and show that it enjoys the finite model property: if a judgement is not valid in the logic, then there is a finite birelational counter-model. Hence, we prove that the logic is decidable.

A modular approach to defining and characterising notions of simulation

Abstract: We propose a modular approach to defining notions of simulation, and modal logics which characterise them. We use coalgebras to model state-based systems, relators to define notions of simulation for such systems, and inductive techniques to define the syntax and semantics of modal logics for coalgebras. We show that the expressiveness of an inductively defined logic for coalgebras w.r.t. a notion of simulation follows from an expressivity condition involving one step in the definition of the logic, and the relator inducing that notion of simulation. Moreover, we show that notions of simulation and associated characterising logics for increasingly complex system types can be derived by lifting the operations used to combine system types, to a relational level as well as to a logical level. We use these results to obtain Baltag's logic for coalgebraic simulation, as well as notions of simulation and associated logics for a large class of non-deterministic and probabilistic systems.