Showing papers on "Modal operator published in 1997"

The Present Perfect as an Epistemic Modal

Abstract: In a number of languages from various language families, the morphology of the present perfect or a form historically derived from the present perfect, expresses a particular evidential category, one that indicates the availability of indirect evidence for the truth of a proposition (the exact interpretation i s discussed in more detai l in the next two sections) . l The phenomenon, to which I give the name PERFECT OF EVIDENTIALITY (PE), i s i l lustrated in ( I ) :

An overview of interpretability logic

Abstract: A miracle happens. In one hand we have a class of marvelously complex theories in predicate logic, theories with 'sufficient coding potential', like PA (Peano Arithmetic) or ZF (Zermelo Fraenkel Set Theory). In the other we have certain modal propositional theories of striking simplicity. We translate the modal operators of the modal theories to certain specic, fixed, defined predicates of the predicate logical theories. These special predicates generally contain an astronomical number of symbols. We interpret the propositional variables by arbitrary predicate logical sentences. And see: the modal theories are sound and complete for this interpretation. They codify precisely the schematic principles in their scope. Miracles do happen ....

Autoepistemic description logics

Abstract: We present Autoepistemic Description Logics (ADLs), in which the language of Description Logics is augmented with modal operators interpreted according to the nonmonotonic logic MKNF. We provide decision procedures for query answering in two very expressive ADLs. We show their representational features by addressing defaults, integrity constraints, role and concept closure. Hence, ADLs provide a formal characterization of a wide variety of nonmonotonic features commonly available in frame-based systems and needed in the development of practical applications.

The Algebraic Structure of Sets of Regions

Abstract: The provision of ontologies for spatial entities is an important topic in spatial information theory. Heyting algebras, co-Heyting algebras, and bi-Heyting algebras are structures having considerable potential for the theoretical basis of these ontologies. This paper gives an introduction to these Heyting structures, and provides evidence of their importance as algebraic theories of sets of regions. The main evidence is a proof that elements of certain Heyting algebras provide models of the Region-Connection Calculus developed by Cohn et al. By using the mathematically well known techniques of “pointless topology”, it is straight-forward to conduct this proof without any need to assume that regions consist of sets of points. Further evidence is provided by a new qualitative theory of regions with indeterminate boundaries. This theory uses modal operators which are related to the algebraic operations present in a bi-Heyting algebra.

Two-Dimensional Modal Logics

Abstract: This chapter contains a technical introduction to the world of multi-dimensional modal logics. We will treat some relatively simple logics with a two-dimensional semantics. In section 2.1, we introduce the family of modal operators we are going to study, with their two-dimensional semantics. In sections 2.2 and 2.3, we study two-dimensional modal logic with unary operators. These sections can be seen as an appetizer for the α-dimensional case which is treated in chapter 5. Section 2.4 deals with the modal logic of composition. This section is an introduction to chapter 3, which is completely devoted to logics with composition as their main connective. Section 2.5, finally, is about two-dimensional tense logic, a subject which is taken up again in chapter 4. We conclude this chapter with some historical notes on the logics described here.

Strong Boethius’ Thesis and Consequential Implication

Abstract: The paper studies the relation between systems of modal logic and systems of consequential implication, a non-material form of implication satisfying “Aristotle's Thesis” (p does not imply not p ) and “Weak Boethius' Thesis” (if p implies q, then p does not imply not q ) Definitions are given of consequential implication in terms of modal operators and of modal operators in terms of consequential implication The modal equivalent of “Strong Boethius' Thesis” (that p implies q implies that p does not imply not q) is identified

Die Modalpartikeln ja, doch und schon : Zu ihrer Syntax, Semantik und Pragmatik

Abstract: The aim of this thesis is to show how modal particles can be integrated into a modular description of language where grammar and pragmatics are seen as two independent but interacting modules. The modular approach is chosen because of the problems connected with a functional approach. On the syntactic level, it is assumed that modal particles are XPs. The fact that modal particles cannot occur alone in the initial field of a clause has, presumably, semantic-pragmatic reasons. It is also argued that the many positions possible for modal particles in a clause can be accounted for by assuming that they are generated in one special position in the (theoretical) syntactic structure, namely as adjuncts to the highest VP, and that other constituents may move past them for information structural reasons that have to do with the interaction of the grammatical focus-background structure and the pragmatic theme-rheme structure. In the light of the differences between the various occurrences of the words ja, doch and schon, modal particles are regarded as a category of their own, separated from categories such as adverbs, focus particles and sentence equivalents. On the semantic level, it is shown that the modular approach makes it possible to minimalistically assume one semantic form per modal particle, irrespective of stress. In the case of the modal particles ja, doch and schon, it is argued that they are non-referential and non-attitudinal. It is suggested that, in view of their common feature, ‘affirmative1, they be analysed as expressions of facticity. They are represented by an operator in a duality group based on the operator FAKT. The meaning is compositionally integrated into the meaning of a clause at the level of semantic form, where it takes scope over the proposition [e INST p]. It interacts with other modal particles and sentence adverbials, the order of which shows the scope relations. Finally, on the pragmatic level, it is maintained that there is no inherent connection between modal particles and the focus-background structure or the theme-rheme structure. Moreover, modal particles do not even seem to be included in the theme-rheme structure. However, they take part in the focus-background structure and may be focussed themselves. The functions of modal particles are derived from their meaning in interaction with the illocutionary force or the sentence mood, the Principle of Relevance, and stress. The functions thus derived are a strengthening or a weakening effect on assertions, the triggering of implicatures, and the indication of the type of relation between the utterance concerned and the context.

Selective µ-calculus: New Modal Operators for Proving Properties on Reduced Transition Systems

Abstract: In model checking for temporal logic, the correctness of a (concurrent) system with respect to a desired behavior is verified by checking whether a structure that models the system satisfies a formula describing the behaviour. Most existing verification techniques, and in particular those defined for concurrent calculi like as CCS, are based on a representation of the concurrent system by means of a labelled transition system. In this approach to verification, state explosion is one of the most serious problems. In this paper we present a new temporal logic, the selective mu-calculus, with the property that only the actions occurring in a formula are relevant to check the formula itself. We prove that the selective mu-calculus is as powerful as the mu-calculus. We define the notion of ρ-bisimulation between transition systems: given a set of actions p,a transition system ρ-bisimulates another one if they have the same behaviour with respect to the actions in p. We prove that, if two transition systems are ρ-equivalent, they preserve all the selective mu-calculus formulae with occurring actions in ρ. Consequently, a formula with occurring actions ρ can be more efficiently checked on a transition system ρ-equivalent to the standard one, but smaller than it.

Prefixed tableaux systems for modal logics with enriched languages

Abstract: We present sound and complete prefixed tableaux systems for various modal logics with enriched languages including the "difference" modal operator [≠] and the "only if" modal operator [--R]. These logics are of special interest in Artificial Intelligence since their expressive power is higher than the standard modal logics and for most of them the satisfiability problem remains decidable. We also include in the paper decision procedures based on these systems. In the conclusion, we relate our work with similar ones from the literature and we propose extensions to other logics.

The Logic of Tune - A Proof-Theoretic Analysis of Intonation

Abstract: This paper presents a proof-theoretic sign-based grammar founded on non-associative non-commutative linear logic which models a compositional theory of the 'information packaging' meaning of intonational contours. Cross-language comparison reveals that in expressing information packaging, different languages exploit word order and prosody in different ways: one single informational construct can be realized by drastically different structural means across languages. Thus for languages such as English and Dutch it can be argued that, roughly speaking, information packaging is structurally realized by means of alternative intonational contours of identical strings, while languages such as Catalan and Turkish have a constant prosodic structure and realize information packaging by means of string order permutations. Such cross-linguistic generalizations suggest that information packaging involves syntax as well as prosody, so that any attempt to reduce informational aspects to either syntax (for Catalan or Turkish) or prosody (for English or Dutch) must be inadequate from a cross-linguistic point of view. The present paper proposes to treat the different structural realizations of information packaging by means of a both intonationally/syntactically and semantically/informationally interpreted sign-based version of the non-associative Lambek calculus, the 'pure logic of residuation'. The signs, the grammatical resources of this formalism, are formmeaning units which reflect the fact that the dimensions of form and meaning contribute to well-formedness in an essentially parallel way. The proof-theoretic categorial engine of the formalism represents phonological head/non-head dependencies in terms of a doubling of the pure logic of residuation which is enriched with unary modal operators, where the unary brackets that come with these operators function as demarcations of specific intonational domains.

Inverses for normal modal operators

Abstract: Given a 1-ary sentence operator ○, we describe L - another 1-ary operator - as as a left inverse of ○ in a given logic if in that logic every formula ϕ is provably equivalent to L○ϕ. Similarly R is a right inverse of ○ if ϕ is always provably equivalent to ○Rϕ. We investigate the behaviour of left and right inverses for ○ taken as the □ operator of various normal modal logics, paying particular attention to the conditions under which these logics are conservatively extended by the addition of such inverses, as well as to the question of when, in such extensions, the inverses behave as normal modal operators in their own right.

Explaining referential/attributive

Abstract: Kaplan, Stalnaker and Wettstein all urge a two-stage theory of language whereon the propositions expressed by sentences are generated prior to being evaluated. A new ambiguity for sentences emerges, propositional rather than syntactic or semantic. Kaplan and Wettstein then propose to explain Donnellan's referential/attributive ambiguity as simply being two-stage propositional ambiguity. This is tacitly seen as further confinnation for twostage theory. Modal ambiguities are prime motivators for two-stage theory, which distinguishes local from exotic evaluation to explain them. But if sentences can be found which exhibit both modal and referential/attributive ambiguity, an apparent paradox arises for a two-stage account. The theory recognizes both singular and general propositions, in Kaplan's senses. But reflecting one sense of such a doubly ambiguous sentence, two-stage theory would seem to need a proposition both singular and general with respect to a definite description attributively used. Since modal operators will come into rendering the problem sentences, an obvious idea is to let scope distinctions rescue two-stage theory from the apparent paradox. But while a rescue based on multiple renderings is proposed, it is not strictly a scope rescue, though different scopes are involved. Readers are asked to trust the author on missing formalities of an intuitively transparent two-sorted modal language that is employed. Two-stage theorists explicitly oppose scope treatments of modal ambiguities seeing them as rivals. Stalnaker, in particular, argues against them. But his arguments are shown not to count against the proposed rescue, on which the anticipated rivalry proves to be minimal.

Extending a Logical Framework with a Modal Connective for Validity

Abstract: Logical frameworks, formal systems for programming consequence relation based proof systems, are well known. The notations that have been proposed are suited best to natural deduction style presentations based on truth consequence. We develop a conservative extension of a typical logical framework providing a modal connective which we can use to formalise validity. We argue that this extension is sensible, and provide example encodings of non-standard logics in its terms.

Reasoning about exceptions

Abstract: In this paper we analyze the conditional logic approach to default logic, the logic that formalizes reasoning about default assumptions. Conditional logic is a popular framework to formalize defeasible reasoning. The conditional sentence if β (the antecedent or condition) then by default α (the consequent or conclusion) is represented in this framework by the formula β > α, where '>' is some kind of implication of conditional logic. In this paper different usages of preference orderings for defeasible conditional logics are discussed. The different usages, so-called minimizing and ordering, are represented by different modal operators. Each operator validates different inference rules. Hence, the combination of different modal operators imposes restrictions on the proof theory of the logic. The restriction discussed in this paper is that a proof rule can be blocked in a derivation due to the fact that another proof rule has already been used earlier in the derivation. We call this the two-phase approach in the proof theory.

Modal Logics of Relations

Abstract: In other chapters we have seen or will see how logics from various origins can be treated in the unifying framework of multi-dimensional modal logic. In this chapter we will apply the same methodology to a very familiar logical system, namely that of first-order logic itself. Our modal analysis of the semantics of first-order logic naturally leads to the multi-dimensional perspective, and within this approach we will encounter some unexpected mysteries such as decidable versions of the predicate calculus.

Finite Bases of Admissible Rules for the Logic S52C

Abstract: We study bimodal logic system S52C having two modal operators □0 and □1, each of which satisfies the axioms of S5 and in addition, an axiom for commutability of modal operators: □0□1p]≡□1□0p. The main result of this paper establishes that the bimodal logic S52C and all its extensions have finite bases for admissible inference rules. We also show that even though the logic S52C is not locally finite, any proper extension of S52C is already locally finite. Moreover, the universal theory of the free algebra of any S52C-logic is decidable. It is shown also that any S52C-logic λ with the adjoined inference rule is structurally complete and that logic has the same set of theorems as the logic λ.