Showing papers on "Modal operator published in 1998"

Products of modal logics, part 1

Abstract: The paper studies many-dimensional modal logics corresponding to products of Kripke frames. It proves results on axiomatisability, the finite model property and decidability for product logics, by applying a rather elaborated modal logic technique: p-morphisms, the finite depth method, normal forms, filtrations. Applications to first order predicate logics are considered too. The introduction and the conclusion contain a discussion of many related results and open problems in the area.

The Acquisition of Modality: Implications for Theories of Semantic Representation

Abstract: The set of English modal verbs is widely recognized to communicate two broad clusters of meanings: epistemic and root modal meanings A number of researchers have claimed that root meanings are acquired earlier than epistemic ones; this claim has subsequently been employed in the linguistics literature as an argument for the position that English modal verbs are polysemous (Sweetser, 1990) In this paper I offer an alternative explanation for the later emergence of epistemic interpret- ations by linking them to the development of the child's theory of mind (Wellman, 1990); if correct, this hypothesis might have important implications for the shape of the semantics of modal verbs It is widely acknowledged in the linguistic literature that modal expressions may be used to communicate at least two broad clusters of meanings: epis- temic modal meanings, which deal with the degree of speaker commitment to the truth of the proposition that forms the complement of the modal, and deontic modal meanings, concerned with the necessity or possibility of acts performed by morally responsible agents, eg obligation and permission (Lyons, 1977; Kratzer, 1981; Coates, 1983; Palmer, 1986, 1990; Sweetser, 1990; Bybee and Fleischman, 1995) The utterances in (1) and (2) (on their preferred interpretations) are examples of epistemic and deontic modality respectively:

Satis ability problem in description logics with modal operators

Abstract: The paper considers the standard concept description language ALC augmented with various kinds of modal operators which can be applied to concepts and axioms. The main aim is to develop methods of proving decidability of the satis ability problem for this language and apply them to description logics with most important temporal and epistemic operators, thereby obtaining satis ability checking algorithms for these logics. We deal with the possible world semantics under the constant domain assumption and show that the expanding and varying domain assumptions are reducible to it. Models with both nite and arbitrary constant domains are investigated. We begin by considering description logics with only one modal operator and then prove a general transfer theorem which makes it possible to lift the obtained results to many systems of polymodal description logic.

Hybridizing concept languages

Abstract: This paper shows how to increase the expressivity of concept languages using a strategy called hybridization. Building on the welldknown correspondences between modal and description logics, two hybrid languages are defined. These languages are called ‘hybrid’ because, as well as the familiar propositional variables and modal operators, they also contain variables across individuals and a binder that binds these variables. As is shown, combining aspects of modal and firstdorder logic in this manner allows the expressivity of concept languages to be boosted in a natural way, making it possible to define number restrictions, collections of individuals, irreflexivity of roles, and TBoxd and ABoxdstatements. Subsequent addition of the universal modality allows the notion of subsumption to be internalized, and enables the representation of queries to arbitrary firstdorder knowledge bases. The paper notes themes shared by the hybrid and concept language literatures, and draws attention to a littledknown body of work by the late Arthur Prior.

Shakespearian modal logic: a labelled treatment of modal identity

Abstract: In this paper we describe a modal proof system arising from the combination of a tableau-like classical system, which incorporates a restricted ("analytical") version of the cut rule, with a label formalism which allows for a specialised, logic dependant unification algorithm. The system provides a uniform proof-theoretical treatment of first-order (normal) modal logics with identity, with and without Barcan formula and/or its converse.

A Modular Presentation of Modal Logics in a Logical Framework

Abstract: We present a theoretical and practical approach to the modular natural deduction presentation of modal logics and their implementation in a logical framework Our work treats a large and well known class of modal logics including K KD T B S S S in a uniform way with respect to soundness and completeness for semantics and faithfulness and adequacy of the implementation Moreover it results in a pleasingly simple and usable implementation of these logics x Introduction Logical Frameworks such as the Edinburgh LF and Isabelle have been proposed as a solution to the problem of the explosion of logics and specialized provers for them However it is also acknowledged that this solution is not perfect these frameworks are best suited for encoding well behaved natural deduction formalisms whose metatheory does not deviate too far from the metatheory of the framework logic Modal logics in particular are considered di cult to implement in a clean direct way e g x and Encodings in both the LF and Isabelle have been proposed see section but they have been either Hilbert style or quite specialized and their correctness is subtle We present a method for encoding a large and useful class of propositional modal logics including K KD T B S S S in a natural deduction setting and show once and for all correctness for every encoding in the class We have implemented our work in Isabelle and the result is a simple usable and completely modular natural deduction implementation of these logics Let us consider in more detail the di culty with modal logics since the problem motivates the approach that we pursue The deduction theorem If by adding A as an axiom we can prove B then we can prove A B without A fails in modal logics A semantic explanation of this is that the standard completeness theorem for modal logics says that A i A is true at every world in every suitable Kripke frame hW Ri where W is the set of worlds and R is the accessibility relation Basically A means w W j w A and the deduction theorem states that w W j w A w W j w B w W j w A B where is implication in the meta language and is implication in the object language But this is false we have only w W j w A j w B w W j w A B Thus a naive embedding of a modal logic in a logical framework captures the wrong conse quence relation One solution to this problem is to turn to Hilbert presentations we reject this as it is well known that they are di cult to use in practice Instead motivated by the above semantic account we take the view of a logic as a Labelled Deductive System LDS proposed by Gabbay among others This approach pairs formulae with labels instead of proving A one proves w A where w represents the current world and w W w A i A Then it becomes possible to give a proof theoretic statement of the deduction theorem which is the analogue of the semantic version The same mechanism yields a direct formalization of modal operators like

A Proof-Theoretic Proof of Functional Completeness for Many Modal and Tense Logics

Abstract: In what follows we shall use display logic to define a proof-theoretic semantics in terms of general introduction schemata. It will be shown that with respect to this semantics the set of connectives {[F], [P], ∧, ¬} is functionally complete for every displayable normal propositional tense logic and the set of connectives {[F], ∧, ¬} is functionally complete for every displayable normal propositional modal logic. It seems that there exists no other proof-theoretic characterization of modal operators (apart from intuitionistic implication ⊃ h ) in the literature.

How much does an agent believe: an extension of modal epistemic logic

Abstract: Modal logics are often criticised for their coarse grain representation of knowledge of possibilities about assertions. That is to say, if two assertions are possible in the current world, their further properties are indistinguishable in the modal formalism even if an agent knows that one of them is true in twice as many possible worlds as compared to the other one. Epistemic logic, that is the logic of knowledge and belief, cannot avoid this shortcomings because it inherits the syntax and semantics of modal logics. In this paper, we develop an extended formalism of modal epistemic logic which will allow an agent to represent its degrees of support about an assertion. The degrees are drawn from qualitative or quantitative dictionaries which are accumulated from agent’s a priori knowledge about the application domain. A possible-world semantics of the logic is developed by using the accessibility hyperelation and the soundness and completeness results are stated. The abstract syntax and semantics are illustrated and motivated by an example from the medical domain.

Extended disjunction and existence properties for some predicate modal logics

Abstract: Extended disjunction and existence properties for predicate modal logics K, K4, T, S4 as well as these logics with the Barcan axiom, and the logic of provability are formulated. The theorem analogous to Harrop’s theorem for these logics is proved.

Questions of modality

Abstract: One goal of linguistic theory is to show which expressions determine which contents in which contexts and, conversely, which contents determine which expressions in which contexts. In order for this goal to be realizable, the concept of frequency-based determinacy form is essential. In this paper, aspects of modal expression, modal contents, and modal contexts are viewed within the framework of determinacy grammar, in particular, modal meaning, modal homonymy and polysemy, modal ambiguity resolution, modal scope and sequencing, and modality as causality. In each case, the adopted approach promises to provide precise answers to the questions raised

Causal action theories and satisfiability planning

Abstract: This dissertation addresses the problem of representing and reasoning about commonsense knowledge of action domains. Until recently, most such work has suppressed the notion of causality, despite its central role in everyday talking and reasoning about actions. There is good reason for this. In general, causality is a difficult notion, both philosophically and mathematically. Nonetheless, it turns out that action representations can be made not only more expressive but also mathematically simpler by representing causality more explicitly. The key is to formalize only a relatively simple kind of causal knowledge: knowledge of the conditions under which facts are caused. In the first part of the dissertation we do this using inference rules and rule-based nonmonotonic formalisms. As we show, an inference rule $\phi\over\psi$ can be understood to represent the knowledge that if $\phi$ is caused then $\psi$ is caused. (Notice that we do not say "$\phi$ causes $\psi$.") This leads to simple and expressive action representations in Reiter's default logic, a rule-based nonmonotonic formalism. This approach also yields action descriptions in logic programming, thus raising the possibility, at least in principle, of automated reasoning about actions and planning. In the second part of the dissertation, we introduce a new modal non-monotonic logic--the logic of "universal causation" (UCL)--specifically designed for describing the conditions under which facts are caused. We show that UCL provides a more traditional semantic account of the mathematically simple approach to causal knowledge that underlies our causal theories of action. For instance, instead of the inference rule $\phi\over\psi$ we write the modal formula $\subset\phi \supset \subset\psi$, where $\subset$ is a modal operator read as "caused." In the third part of the dissertation, we show that a subset of UCL is well-suited for automated reasoning about actions. In particular, we show that the class of "simple" UCL theories provides an expressive basis for the computationally challenging task of automated planning. Simple UCL theories have a concise translation into classical logic, and, as we show, the classical models of the translation correspond to valid plans. This enables "satisfiability planning" with causal action theories, with "state of the art" performance on large classical planning problems.

A Note on Classical Modal Relevant Algebras.

Abstract: R. Meyer and E. Mares [10] studied the logic CNR, a classical relevant logic with a modal connective. Such logic is of type S4. As it has been noted in [10], the addition of a modal operator in the language of classical relevant logic and the axioms φ ∧ ψ → (φ ∧ ψ) and (φ ∧ ψ) → ( φ ∧ ψ) do not necessarily produce theorems such that as (φ→ ψ) → φ → ψ or (♦φ→ ψ) → (φ→ ψ). These formulas are well known theorems of classical modal logic [9]. We will define here classical relevant algebras with a modal operator. We shall investigate some types of algebras

Basic logic for ontic and deontic modalities

Abstract: The difficulty to interpret the iteration of modalities, already ontic and still more deontic, incites to pay attention to the system B of basic modal logic that John L. Pollock proposed in 1967. The Pollock’s system brought all the theses which, in the classical ontic modal systems, from Sl to S5, contain no iteration of the modal functors. With this basic ontic system we characterize a basic deontic system, and a basic ontico-deontic system, the former including all the theses of the first two. Each of the three systems is based axiomatically and assorted with a semantics for which the soundness and the completeness hold. In [6] John L. Pollock proposed a basic modal logic, B, whose main characteristics was that it did not admit any iterated modality, i.e. no modal operator would appear within the scope of another modality. The system B 1 may be based on two axioms: A1 Lp ⊃ p A2 L(p ⊃ q) ⊃ (Lp ⊃ Lq) and the four inference rules: R1 Rule of substitution, proviso that the substitution of a WFF (well formed formula) does not generate any iteration of modality R2 Rule of replacement, with the classical definitions of PC (propositional calculus) and the definition