Showing papers on "Modal operator published in 2008"

Contextual modal type theory

Abstract: The intuitionistic modal logic of necessity is based on the judgmental notion of categorical truth. In this article we investigate the consequences of relativizing these concepts to explicitly specified contexts. We obtain contextual modal logic and its type-theoretic analogue. Contextual modal type theory provides an elegant, uniform foundation for understanding metavariables and explicit substitutions. We sketch some applications in functional programming and logical frameworks.

'knowable' as 'known after an announcement' ∗

Abstract: Public announcement logic is an extension of multi-agent epistemic logic with dynamic operators to model the informational consequences of announcements to the entire group of agents. We propose an extension of public announcement logic with a dynamic modal operator that expresses what is true after any announcement: ♦ϕ expresses that there is a truthful announcement ψ after which ϕ is true. This logic gives a perspective on Fitch’s knowability issues: for which formulas ϕ does it hold that ϕ → ♦Kϕ? We give various semantic results, and we show completeness for a Hilbert-style axiomatization of this logic. There is a natural generalization to a logic for arbitrary events.

Epistemic logic and information update

Abstract: Epistemic logic investigates what agents know or believe about certain factual descriptions of the world, and about each other. It builds on a model of what information is (statically) available in a given system, and isolates general principles concerning knowledge and belief. The information in a system may well change as a result of various changes: events from the outside, observations by the agents, communication between the agents, etc. This requires information updates. These have been investigated in computer science via interpreted systems ; in philosophy and in artificial intelligence their study leads to the area of belief revision. A more recent development is called dynamic epistemic logic. Dynamic epistemic logic is an extension of epistemic logic with dynamic modal operators for belief change (i.e., information update). It is the focus of our contribution, but its relation to other ways to model dynamics will also be discussed in some detail.

A modal deconstruction of access control logics

Abstract: We present a translation from a logic of access control with a "says" operator to the classical modal logic S4. We prove that the translation is sound and complete. We also show that it extends to logics with boolean combinations of principals and with a "speaks for" relation. While a straightforward definition of this relation requires second-order quantifiers, we use our translation for obtaining alternative, quantifier-free presentations. We also derive decidability and complexity results for the logics of access control.

Decidable and Undecidable Fragments of Halpern and Shoham's Interval Temporal Logic: Towards a Complete Classification

Abstract: Interval temporal logics are based on temporal structures where time intervals, rather than time instants, are the primitive ontological entities. They employ modal operators corresponding to various relations between intervals, known as Allen's relations. Technically, validity in interval temporal logics translates to dyadic second-order logic, thus explaining their complex computational behavior. The full modal logic of Allen's relations, called HS, has been proved to be undecidable by Halpern and Shoham under very weak assumptions on the class of interval structures, and this result was discouraging attempts for practical applications and further research in the field. A renewed interest has been recently stimulated by the discovery of interesting decidable fragments of HS. This paper contributes to the characterization of the boundary between decidability and undecidability of HS fragments. It summarizes known positive and negative results, it describes the main techniques applied so far in both directions, and it establishes a number of new undecidability results for relatively small fragments of HS.

Many-valued modal logics: a simple approach

Abstract: 1.1 In standard modal logics, the worlds are 2-valued in the following sense: there are 2 values (true and false) that a sentence may take at a world. Technically, however, there is no reason why this has to be the case. The worlds could be many-valued. This paper presents one simple approach to a major family of many-valued modal logics, together with an illustration of why this family is philosophically interesting.

Undecidability of the unification and admissibility problems for modal and description logics

Abstract: We show that the unification problem “is there a substitution instance of a given formula that is provable in a given logicq” is undecidable for basic modal logics K and K4 extended with the universal modality. It follows that the admissibility problem for inference rules is undecidable for these logics as well. These are the first examples of standard decidable modal logics for which the unification and admissibility problems are undecidable. We also prove undecidability of the unification and admissibility problems for K and K4 with at least two modal operators and nominals (instead of the universal modality), thereby showing that these problems are undecidable for basic hybrid logics. Recently, unification has been introduced as an important reasoning service for description logics. The undecidability proof for K with nominals can be used to show the undecidability of unification for Boolean description logics with nominals (such as ALCO and SHIQO). The undecidability proof for K with the universal modality can be used to show that the unification problem relative to role boxes is undecidable for Boolean description logics with transitive roles, inverse roles, and role hierarchies (such as SHI and SHIQ).

Logical Extensions of Aristotle's Square

Abstract: We start from the geometrical-logical extension of Aristotle’s square in [6,15] and [14], and study them from both syntactic and semantic points of view. Recall that Aristotle’s square under its modal form has the following four vertices: A is □α, E is $$\square eg\alpha$$ , I is $$ eg\square eg\alpha$$ and O is $$\square eg\alpha$$ , where α is a logical formula and □ is a modality which can be defined axiomatically within a particular logic known as S5 (classical or intuitionistic, depending on whether $$ eg$$ is involutive or not) modal logic. [3] has proposed extensions which can be interpreted respectively within paraconsistent and paracomplete logical frameworks. [15] has shown that these extensions are subfigures of a tetraicosahedron whose vertices are actually obtained by closure of $$\{\alpha,\square\alpha\}$$ by the logical operations $$\{ eg,\wedge,\vee\}$$ , under the assumption of classical S5 modal logic. We pursue these researches on the geometrical-logical extensions of Aristotle’s square: first we list all modal squares of opposition. We show that if the vertices of that geometrical figure are logical formulae and if the sub-alternation edges are interpreted as logical implication relations, then the underlying logic is none other than classical logic. Then we consider a higher-order extension introduced by [14], and we show that the same tetraicosahedron plays a key role when additional modal operators are introduced. Finally we discuss the relation between the logic underlying these extensions and the resulting geometrical-logical figures.

Discrete Dualities for Heyting Algebras with Operators

Abstract: Discrete dualities are presented for Heyting algebras with various modal operators, for Heyting algebras with an external negation, for symmetric Heyting algebras, and for Heyting-Brouwer algebras.

Dynamic epistemic logic with justification

Abstract: Justification Logic is the study of a family of logics used to reason about justified true belief. Dynamic Epistemic Logic is the study of logics used to reason about communication and true belief. This dissertation is a first step in merging these two areas, in that it defines theories for a joint language in which we may reason about communication alongside justified true belief. After some preliminary matters, we go through a comprehensive survey of Dynamic Epistemic Logic, which primes us for the work at the end of the text. We then move into the core work of the dissertation, where we introduce a number of extensions of existing languages and theories of Justification Logic. Our extensions are all based on the work of Sergei Artemov, who extended the language of propositional logic by the addition of formula-labeling terms. This extension allows us to take a term t and a formula p and form the new formula t:p. The terms have a derivation-compatible structure that allows us to view terms as evidence verifying the truth of the formulas they label, which provides us with a means for reasoning about justified true belief. We look at extensions of these theories that allow us to reason about evidence admissibility: the new formula t G p lets us express that t is admissible as evidence for p, by which we mean that t may be taken into account when considering the truth of p, though t need not conclusively validate p. A further extension adds a unary modal operator s that we use to reason about alternative evidence possibilities. Nominated extensions of the latter languages allow us to express a notion of dynamic evidence introduction, whereby we may introduce a term t as admissible as evidence for p. These extensions lead us to the final chapter of the text, where we combine our various systems of Justification Logic with the framework of Dynamic Epistemic Logic. Such joint theories contribute to the ongoing work aiming to provide a better foundational account of the reasoning of computational social agents.

A Topological Study of Contextuality and Modality in Quantum Mechanics

Abstract: Kochen–Specker theorem rules out the non-contextual assignment of values to physical magnitudes Here we enrich the usual orthomodular structure of quantum mechanical propositions with modal operators This enlargement allows to refer consistently to actual and possible properties of the system By means of a topological argument, more precisely in terms of the existence of sections of sheaves, we give an extended version of Kochen–Specker theorem over this new structure This allows us to prove that contextuality remains a central feature even in the enriched propositional system

Modal Logic: An Introduction to its Syntax and Semantics

Abstract: In this text, a variety of modal logics at the sentential, first-order, and second-order levels are developed with clarity, precision and philosophical insight. All of the S1-S5 modal logics of Lewis and Langford, among others, are constructed. A matrix, or many-valued semantics, for sentential modal logic is formalized, and an important result that no finite matrix can characterize any of the standard modal logics is proven. Exercises, some of which show independence results, help to develop logical skills. A separate sentential modal logic of logical necessity in logical atomism is also constructed and shown to be complete and decidable. On the first-order level of the logic of logical necessity, the modal thesis of anti-essentialism is valid and every de re sentence is provably equivalent to a de dicto sentence. An elegant extension of the standard sentential modal logics into several first-order modal logics is developed. Both a first-order modal logic for possibilism containing actualism as a proper part as well as a separate modal logic for actualism alone are constructed for a variety of modal systems. Exercises on this level show the connections between modal laws and quantifier logic regarding generalization into, or out of, modal contexts and the conditions required for the necessity of identity and non-identity. Two types of second-order modal logics, one possibilist and the other actualist, are developed based on a distinction between existence-entailing concepts and concepts in general. The result is a deeper second-order analysis of possibilism and actualism as ontological frameworks. Exercises regarding second-order predicate quantifiers clarify the distinction between existence-entailing concepts and concepts in general. Modal Logic is ideally suited as a core text for graduate and undergraduate courses in modal logic, and as supplementary reading in courses on mathematical logic, formal ontology, and artificial intelligence

A temporal logic for Markov chains

Abstract: Most models of agents and multi-agent systems include information about possible states of the system (that defines relations between states and their external characteristics), and information about relationships between states. Qualitative models of this kind assign no numerical measures to these relationships. At the same time, quantitative models assume that the relationships are measurable, and provide numerical information about the degrees of relations. In this paper, we explore the analogies between some qualitative and quantitative models of agents/processes, especially those between transition systems and Markovian models.Typical analysis of Markovian models of processes refers only to the expected utility that can be obtained by the process. On the other hand, modal logic offers a systematic method of describing phenomena by combining various modal operators. Here, we try to exploit linguistic features, offered by propositional modal logic, for analysis of Markov chains and Markov decision processes. To this end, we propose Markov temporal logic - a multi-valued logic that extends the branching time logic CTL*.

Modal Semirings Revisited

Abstract: A new axiomatisation for domain and codomain on semirings and Kleene algebras is proposed. It is simpler, more general and more flexible than a predecessor, and it is particularly suitable for program analysis and construction via automated deduction. Different algebras of domain elements for distributive lattices, (co-)Heyting algebras and Boolean algebras arise by adapting this axiomatisation. Modal operators over all these domain algebras can then easily be defined. The calculus of the previous axiomatisation arises as a special case. An application in terms of a fully automated proof of a modal correspondence result for Lob's formula is also presented.

A Tableau Decision Algorithm for Dynamic Description Logic

Abstract: By introducing a dynamic dimension into the description logic,two kinds of dynamic description logics have been proposed in the literatures for representing and reasoning about knowledge of dynamic application domainsBut both of them lack efficient decision algorithmsThis paper presents a Tableau decision algorithm for the dynamic description logic D-ALCOD-ALCO is a combination of the description logic ALCO,the dynamic logic,and an action theory based on the possible models approachIn D-ALCO,based on domain ontologies expressed in ALCO,atomic actions are described by specifying their preconditions and effects;With the help of standard action constructors of the dynamic logic,complex actions can be described also;Both atomic actions and complex actions are then used as modal operators in the construction of formulasThe Tableau decision algorithm for D-ALCO forms an elaborated combination of the Tableau algorithm for ALCO,the decision algorithm for propositional dynamic logic,and the embodiment of the possible models approachThe authors prove that the algorithm is terminating,sound and complete for deciding the satisfiability of D-ALCO formulas with the Open World AssumptionThe algorithm is flexible and can be extended for more expressive dynamic description logics such as the D-ALCQO and the D-ALCQIO

A Completeness Proof for an Infinitary Tense Logic

Abstract: In his forthcoming examination of G. H. von Wright's tense-logic [4], Krister Segerberg studies certain infinitary extensions of the original tense-logic created by von Wright. For one of these extensions the completeness problem turned out to be harder than was expected at first sight. This paper is devoted to a proof of a completeness theorem for the extension in question, called Wl by Segerberg. We use a countable language of ordinary prepositional logic supplied with two modal operators: O ("tomorrow") and D ("always"). The relevant semantics for tense-logic based on this language uses the frame 9^ = , where the successorrelation is the accessibility-relation for O and n A is true at k. We assume that the reader is familiar with ordinary Kripkesemantics for modal languages and, in particular, that he understands what it means that "9ft is a model on the frame 9^". We shall use O(A] as a shorthand for

Exploring a Syntactic Notion of Modal Many-Valued Logics

Abstract: We propose a general semantic notion of modal many-valued logic. Then, we explore the diculties to characterize this notion in a syntactic way and analyze the existing literature with respect to this framework.

Modal Epistemology: Our Knowledge of Necessity and Possibility

Abstract: I survey a number of views about how we can obtain knowledge of modal propositions, propositions about necessity and possibility. One major approach is that whether a proposition or state of affairs is conceivable tells us something about whether it is possible. I examine two quite different positions that fall under this rubric, those of Yablo and Chalmers. One problem for this approach is the existence of necessary a posteriori truths and I deal with some of the ways in which these authors respond to the problem, including the use of two-dimensional modal semantics. Conventionalism about modality offers a complementary approach to modal epistemology, prompting us to identify our knowledge of modal truths with our mastery of linguistic or conceptual conventions. Finally, I discuss an approach to modal epistemology deriving from David Lewis's work that seeks to identify structural features of the modal space over which necessity and possibility are defined.

Man Muss Immer Umkehren

Abstract: The 19th century geometrist Jacobi famously said that one should always try to invert every geometrical theorem. But his advice applies much more widely! Choose any class of relational frames, and you can study its valid modal axioms. But now turn the perspective around, and fix some modal axiom beforehand. You can then find the class of frames where the axiom is guaranteed to hold by 'modal correspondence' analysis ‐ and we all know the famous examples of that. It may look as if this style of analysis is tied to one particular semantics, say relational frames: but it is not. Correspondence analysis also works on neighbourhood models, telling us, e.g., just which modal axioms collapse these to binary relational frames. We will show how this same style of inverse thinking also applies to modern dynamic logics of information change. Basic axioms for knowledge after information update !A tell us what sort of operation must be used for updating a given model M to a new one incorporating A. Likewise, we will show how modal axioms for (conditional) beliefs that hold after revision actions *A actually fix one particular operation of changing the relative plausibility orderings which agents have on the universe of possible worlds. And finally, going back to the traditional heartland of logic, we show how we can read standard predicate-logical axioms as constraints on the sort of abstract 'process models' that lie at the heart of first-order semantics, properly understood. In all these cases, in order for the inversion to work and illuminate a given subject, we need to step back and reconsider our standard modeling. But that, I think, is what Shahid Rahman is all about. 2 Standard modal frame correspondences One of the most attractive features of the semantics of modal logic is the match between modal axioms and corresponding patterns in the accessibility relation between worlds. This can be seen by giving a class of models, say temporal or epistemic, and then axiomatizing its set of modal validities. On top of the minimal modal logic which holds under all circumstances, one gets additional axioms reflecting more specific structure. For general background to modal completeness theory, as well as the rest of this paper, we refer to the Handbook of Modal Logic (P. Blackburn, J. van Benthem & F. Wolter, eds.) which has just come out with Elsevier Science Publishers, Amsterdam, 1997.

Contextual Epistemic Logic

Abstract: One of the highlights of recent informal epistemology is its growing theoretical emphasis upon various notions of context. The present paper addresses the connections between knowledge and context within a formal approach. To this end, a "contextual epistemic logic", CEL, is proposed, which consists of an extension of standard S5 epistemic modal logic with appropriate reduction axioms to deal with an extra contextual operator. We describe the axiomatics and supply both a Kripkean and a dialogical semantics for CEL. An illustration of how it may fruitfully be applied to informal epistemological matters is provided.

On Extending RuleML for Modal Defeasible Logic

Abstract: In this paper we present a general methodology to extend Defeasible Logic with modal operators. We motivate the reasons for this type of extension and we argue that the extension will allow for a robust knowledge framework in different application areas. The paper presents an extension of RuleML to capture Modal Defeasible Logic.

A note on assumption-completeness in modal logic

Abstract: We study the notion of assumption-completeness, which is a property of belief models first introduced in [18]. In that paper it is considered a limitative result - of significance for game theory - if a given language does not have an assumption-complete belief model. We show that there are assumption-complete models for the basic modal language (Theorem 8).

Intuition and Modal Error

Abstract: Modal intuitions are not only the primary source of modal knowledge but also the primary source of modal error. An explanation of how modal error arises—and, in particular, how erroneous modal intuitions arise—is an essential part of a comprehensive theory of knowledge and evidence. But, more than that, such an explanation is essential to identifying and eliminating modal errors in our day-to-day philosophical practice. According to the theory of modal error given here, modal intuitions retain their evidential force in spite of their fallibility, and erroneous modal intuitions turn out to be in principle identifiable and eliminable by subjecting intuitions to properly conducted a priori dialectic and theory construction. And, thus, the classical method of intuition-driven philosophical investigation is exonerated. I begin with a summary of certain preliminaries: the phenomenology of intuitions, their fallibility, the nature of concept-understanding and its relationship to the reliability of intuitions, and so forth. This is followed by an inventory of standard sources of modal error. I then go on to discuss two specific sources: the first has to do with the failure to distinguish between metaphysical possibility and various kinds of epistemic possibility; the second, with the local misunderstanding of one’s concepts (as opposed to out-and-out misunderstanding, as in Burge’s original arthritis case). The first source of error,

Modal-type orthomodular logic

Abstract: In this paper we enrich the orthomodular structure by adding a modal operator, following a physical motivation. A logical system is developed, obtaining algebraic completeness and completeness with respect to a Kripke-style semantic founded on Baer *-semigroups as in [20].

Knowledge Assessment: A Modal Logic Approach

Abstract: The possible worlds semantics is a fruitful approach used in Artificial Intelligence (AI) for both modelling as well as reasoning about knowledge in agent systems via modal logics. In this work our main idea is not to model/reason about knowledge but to provide a theoretical framework for knowledge assessment (KA) with the help of Monatague-Scott (MS) semantics of modal logic. In KA questions asked and answers collected are the central elements and knowledge notions will be defined from these (i.e., possible states of knowledge of subjects in a population with respect to a field of information).

Extending a Defeasible Reasoner with Modal and Deontic Logic Operators

Abstract: Defeasible logic is a non-monotonic formalism that deals with incomplete and conflicting information. Modal logic deals with necessity and possibility, exhibiting defeasibility; thus, it is possible to combine defeasible logic with modal operators. This paper reports on the extension of the DR-DEVICE defeasible reasoner with modal and deontic logic operators. The aim is a practical defeasible reasoner that will take advantage of the expressiveness of modal logics and the flexibility to define diverse agent types and behaviors.