Coinductive models and normal forms for modal logics (or how we learned to stop worrying and love coinduction)

We present a coinductive definition of models for modal logics and show that it provides a homogeneous framework in which it is possible to include different modal languages ranging from classical modalities to operators from hybrid and memory logics. Moreover, results that had to be proved separately for each different language—but whose proofs were known to be mere routine—now can be proved in a general way. We show, for example, that we can have a unique definition of bisimulation for all these languages, and prove a single invariance-under-bisimulation theorem.We then use the new framework to investigate normal forms for modal logics. The normal form we introduce may have a smaller modal depth than the original formula, and it is inspired by global modalities like the universal modality and the satisfiability operator from hybrid logics. These modalities can be extracted from under the scope of other operators. We provide a general definition of extractable modalities and show how to compute extracted normal forms. As it is the case with other classical normal forms—e.g., the conjunctive normal form of propositional logic—the extracted normal form of a formula can be exponentially bigger than the original formula, if we require the two formulas to be equivalent. If we only require equi-satisfiability, then every modal formula has an extracted normal form which is only polynomially bigger than the original formula, and it can be computed in polynomial time.     

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

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.     

The Logic of Location

I consider the idea of a propositional logic of location based on the following semantic framework, derived from ideas of Prior. We have a collection L of locations and a collection S of statements such that a statement may be evaluated for truth at each location. Typically one and the same statement may be true at one location and false at another. Given this semantic framework we may proceed in two ways: introducing names for locations, predicates for the relations among them and an “at” preposition to express the value of statements at locations; or introduce statement operators which do not name locations but whose truth-conditional effect depends on the truth or falsity of embedded statements at various locations. The latter is akin to Prior’s approach to tense logic. In any logic of location there will be some basic operators which we can define. By ringing the changes on the topology of locations, different logical systems may be generated, and the challenge for the logician is then in each case to find operators, axioms and rules yielding a proof theory adequate to the semantics. The generality of the approach is illustrated with familiar and not so familiar examples from modal, tense and place logic, mathematics, and even the logic of games.

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Models for Counterparts

Lewis proposed to test the validity of a modal thesis by checking whether its possible-world translation is a theorem of counterpart theory. However, that criterion fails to validate many standard modal laws, thus raising doubts about the logical adequacy of the Lewisian framework. The present paper considers systems of counterpart theory of increasing strength and shows how each can be motivated by exhibiting a suitable intended model. In particular, perfect counterpart theory validates all the desired modal laws and therefore provides a way out of the logical objection. Finally, a weakening of perfect counterpart theory is put forward as a response to some metaphysical objections.     

Formal systems for modal operators on locales

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 Monodic Fragment of Propositional Term Modal Logic

We study term modal logics, where modalities can be indexed by variables that can be quantified over. We suggest that these logics are appropriate for reasoning about systems of unboundedly many reasoners and define a notion of bisimulation which preserves propositional fragment of term modal logics. Also we show that the propositional fragment is already undecidable but that its monodic fragment (formulas using only one free variable in the scope of a modality) is decidable, and expressive enough to include interesting assertions.

     

On Modal Logics Characterized by Models with Relative Accessibility Relations: Part II

This work is divided in two papers (Part I and Part II). In Part I, we introduced the class of Rare-logics for which the set of terms indexing the modal operators are hierarchized in two levels: the set of Boolean terms and the set of terms built upon the set of Boolean terms. By investigating different algebraic properties satisfied by the models of the Rare-logics, reductions for decidability were established by faithfully translating the Rare-logics into more standard modal logics (some of them contain the universal modal operator).In Part II, we push forward the results from Part I. For Rare-logics with nominals (present at the level of formulae and at the level of modal expressions), we show that the constructions from Part I can be extended although it is technically more involved. We also characterize a class of standard modal logics for which the universal modal operator can be eliminated as far as satifiability is concerned. Although the previous results have a semantic flavour, we are also able to define proof systems for Rare-logics from existing proof systems for the corresponding standard modal logics. Last, but not least, decidability results for Rare-logics are established uniformly, in particular for information logics derived from rough set theory.Since this paper is the continuation of Part I, we do not recall here the definitions of Part I although we refer to them.     

Generalizing Deontic Action Logic

We introduce a multimodal framework of deontic action logic which encodes the interaction between two fundamental procedures in normative reasoning: conceptual classification and deontic classification. The expressive power of the framework is noteworthy, since it combines insights from agency logic and dynamic logic, allowing for a representation of many kinds of normative conflicts. We provide a semantic characterization for three axiomatic systems of increasing strength, showing how our approach can be modularly extended in order to get different levels of analysis of normative reasoning. Finally, we discuss ways in which the framework can be used to capture other formalisms proposed in the literature, as well as to model searching problems in Artificial Intelligence.

     

A Logical Account of Formal Argumentation

In the current paper, we re-examine how abstract argumentation can be formulated in terms of labellings, and how the resulting theory can be applied in the field of modal logic. In particular, we are able to express the (complete) extensions of an argumentation framework as models of a set of modal logic formulas that represents the argumentation framework. Using this approach, it becomes possible to define the grounded extension in terms of modal logic entailment.     

On Logics with Coimplication

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 Gödel-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.     

The Logic of “Most” and “Mostly”

The paper suggests a modal predicate logic that deals with classical quantification and modalities as well as intermediate operators, like “most” and “mostly”. Following up the theory of generalized quantifiers, we will understand them as two-placed operators and call them determiners. Quantifiers as well as modal operators will be constructed from them. Besides the classical deduction, we discuss a weaker probabilistic inference “therefore, probably” defined by symmetrical probability measures in Carnap’s style. The given probabilistic inference relates intermediate quantification to singular statements: “Most S are P” does not logically entail that a particular individual S is also P, but it follows that this is probably the case, where the probability is not ascribed to the propositions but to the inference. We show how this system deals with single case expectations while predictions of statistical statements remain generally problematic.     

Denotation of contextual modal type theory (CMTT): Syntax and meta-programming

The modal logic S4 can be used via a Curry–Howard style correspondence to obtain a λ-calculus. Modal (boxed) types are intuitively interpreted as ‘closed syntax of the calculus’. This λ-calculus is called modal type theory—this is the basic case of a more general contextual modal type theory, or CMTT.CMTT has never been given a denotational semantics in which modal types are given denotation as closed syntax. We show how this can indeed be done, with a twist. We also use the denotation to prove some properties of the system.     

The epistemological objection to modal primitivism

Modal primitivists hold that some modal truths are primitively true. They thus seem to face a special epistemological problem: how can primitive modal truths be known? The epistemological objection has not been adequately developed in the literature. I undertake to develop the objection, and then to argue that the best formulation of the epistemological objection targets all realists about modality, rather than the primitivist alone. Furthermore, the moves available to reductionists in response to the objection are also available to primitivists. I conclude by suggesting that extant theories of the epistemology of modality are not sensitive to the question of primitivism versus reductionism.

     

Term-Modal Logics

Many powerful logics exist today for reasoning about multi-agent systems, but in most of these it is hard to reason about an infinite or indeterminate number of agents. Also the naming schemes used in the logics often lack expressiveness to name agents in an intuitive way.To obtain a more expressive language for multi-agent reasoning and a better naming scheme for agents, we introduce a family of logics called term-modal logics. A main feature of our logics is the use of modal operators indexed by the terms of the logics. Thus, one can quantify over variables occurring in modal operators. In term-modal logics agents can be represented by terms, and knowledge of agents is expressed with formulas within the scope of modal operators.This gives us a flexible and uniform language for reasoning about the agents themselves and their knowledge. This article gives examples of the expressiveness of the languages and provides sequent-style and tableau-based proof systems for the logics. Furthermore we give proofs of soundness and completeness with respect to the possible world semantics.     

How to Revise a Total Preorder

Most approaches to iterated belief revision are accompanied by some motivation for the use of the proposed revision operator (or family of operators), and typically encode enough information in the epistemic state of an agent for uniquely determining one-step revision. But in those approaches describing a family of operators there is usually little indication of how to proceed uniquely after the first revision step. In this paper we contribute towards addressing that deficiency by providing a formal framework which goes beyond the first revision step in two ways. First, the framework is obtained by enriching the epistemic state of an agent starting from the following intuitive idea: we associate to each world x two abstract objects x ⁺ and x ⁻, and we assume that, in addition to preferences over the set of worlds, we are given preferences over this set of objects as well. The latter can be considered as meta-information encoded in the epistemic state which enables us to go beyond the first revision step of the revision operator being applied, and to obtain a unique set of preferences over worlds. We then extend this framework to consider, not only the revision of preferences over worlds, but also the revision of this extended structure itself. We look at some desirable properties for revising the structure and prove the consistency of these properties by giving a concrete operator satisfying all of them. Perhaps more importantly, we show that this framework has strong connections with two other types of constructions in related areas. Firstly, it can be seen as a special case of preference aggregation which opens up the possibility of extending the framework presented here into a full-fledged framework for preference aggregation and social choice theory. Secondly, it is related to existing work on the use of interval orderings in a number of different contexts.     

Models for normal intuitionistic modal logics

Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given for analogues of the modal systemK based on Heyting's prepositional logic. It is shown that these two relations can combine with each other in various ways. Soundness and completeness are proved for systems with only the necessity operator, or only the possibility operator, or both. Embeddings in modal systems with several modal operators, based on classical propositional logic, are also considered. This paper lays the ground for an investigation of intuitionistic analogues of systems stronger thanK. A brief survey is given of the existing literature on intuitionistic modal logic.     

Proof-Theoretic Functional Completeness for the Hybrid Logics of Everywhere and Elsewhere

A hybrid logic is obtained by adding to an ordinary modal logic further expressive power in the form of a second sort of propositional symbols called nominals and by adding so-called satisfaction operators. In this paper we consider hybridized versions of S5 (“the logic of everywhere”) and the modal logic of inequality (“the logic of elsewhere”). We give natural deduction systems for the logics and we prove functional completeness results.     

Understanding adolescent type 1 diabetes self-management as an adaptive process: A grounded theory approach

Objective: To develop a conceptual understanding of the process of adapting to the self-management of type 1 diabetes during adolescence.

Design: Participants were recruited from a National Health Service paediatric diabetes service within the south-west of England which runs six countywide diabetes clinics. Thirteen interviews were conducted using a social constructivist grounded theory approach.

Results: The findings illustrate how self-management can be understood in terms of a continuum-based framework, ranging from difficulties with, to successful self-management. Adaptation within the continuum can further be understood by specific transitional phases and process mechanisms, providing further depth to individuals’ experiences of adaptation.

Conclusion: This investigation provides a conceptual understanding of the complex issues adolescents encounter while adapting to and integrating a diabetes self-management regime into their lives. It provides an invaluable framework for exploring psychological mechanisms and contextualising them within a self-management continuum. Implications for healthcare professionals are discussed and further research proposes whether the model could be applicable to other chronic illnesses.     

Game logic and its applications I

This paper provides a logic framework for investigations of game theoretical problems. We adopt an infinitary extension of classical predicate logic as the base logic of the framework. The reason for an infinitary extension is to express the common knowledge concept explicitly. Depending upon the choice of axioms on the knowledge operators, there is a hierarchy of logics. The limit case is an infinitary predicate extension of modal propositional logic KD4, and is of special interest in applications. In Part I, we develop the basic framework, and show some applications: an epistemic axiomatization of Nash equilibrium and formal undecidability on the playability of a game. To show the formal undecidability, we use a term existence theorem, which will be proved in Part II.The authors thank Hiroakira Ono for helpful discussions and encouragements from the early stage of this research project, and Philippe Mongin, Mitio Takano and a referee of this journal for comments on earlier versions of this paper. The first and second authors are partially supported, respectively, by Tokyo Center of Economic Research and Grant-in-Aids for Scientific Research 04640215, Ministry of Education, Science and Culture.Presented by H. Ono