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16
Modal Logic: A Semantic Perspective
 ETHICS
, 1988
"... This chapter introduces modal logic as a tool for talking about graphs, or to use more traditional terminology, as a tool for talking about Kripke models and frames. We want the reader to gain an intuitive appreciation of this perspective, and a firm grasp of the key technical ideas (such as bisimul ..."
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This chapter introduces modal logic as a tool for talking about graphs, or to use more traditional terminology, as a tool for talking about Kripke models and frames. We want the reader to gain an intuitive appreciation of this perspective, and a firm grasp of the key technical ideas (such as bisimulations) which underly it. We introduce the syntax and semantics of basic modal logic, discuss its expressivity at the level of models, examine its computational properties, and then consider what it can say at the level of frames. We then move beyond the basic modal language, examine the kinds of expressivity offered by a number of richer modal logics, and try to pin down what it is that makes them all ‘modal’. We conclude by discussing an example which brings many of the ideas we discuss into play: games.
Definability of Polyadic Lifts of Generalized Quantifiers
 Journal of Logic, Language and Information
, 1999
"... We study generalized quantifiers on finite structures. With every function f : ! ! ! we associate a quantifier Q f by letting Q f x' say "there are at least f(n) elements x satisfying ' , where n is the size of the universe". This is the general form of what is known as a mono ..."
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We study generalized quantifiers on finite structures. With every function f : ! ! ! we associate a quantifier Q f by letting Q f x' say "there are at least f(n) elements x satisfying ' , where n is the size of the universe". This is the general form of what is known as a monotone quantifier of type h1i . We study so called polyadic lifts of such quantifiers. The particular lifts we consider are Ramseyfication, branching and resumption. In each case we get exact criteria for definability of the lift in terms of simpler quantifiers. 1 Introduction and preliminaries Monadic generalized quantifiers express properties of isomorphism types of monadic structures, or equivalently, relations between cardinal numbers. Arbitrary (polyadic) generalized quantifiers, however, express properties of isomorphism types of arbitrary relational structures. Apart from the issue of the adequacy of the term `quantifier ' here  only monadic generalized quantifiers deal with quantities  the step...
A Lindström theorem for modal logic
 Modal Logic and Process Algebra
, 1995
"... A modal analogue of Lindstrom's characterization of firstorder logic is proved. Basic modal logics are characterized as the only modal logics that have a notion of finite rank, or, equivalently, as the strongest modal logic whose formulas are preserved under ultraproducts over !. Also, bas ..."
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A modal analogue of Lindstrom's characterization of firstorder logic is proved. Basic modal logics are characterized as the only modal logics that have a notion of finite rank, or, equivalently, as the strongest modal logic whose formulas are preserved under ultraproducts over !. Also, basic modal logic is the strongest classical logic whose formulas are preserved under bisimulations and ultraproducts over !.
Basic Quantifier Theory
"... ing from the domain of discourse, we can say that determiner interpretations (henceforth: determiners) pick out binary relations on sets of individuals, on arbitrary universes (or: domains of discourse) E. Notation: DEAB. We call A the restriction of the quantifier and B its body. If DEAB is the tra ..."
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ing from the domain of discourse, we can say that determiner interpretations (henceforth: determiners) pick out binary relations on sets of individuals, on arbitrary universes (or: domains of discourse) E. Notation: DEAB. We call A the restriction of the quantifier and B its body. If DEAB is the translation of a simple sentence consisting of a quantified noun phrase with an intransitive verb phrase then the noun denotation is the restriction and the verb phrase denotation the body. See figure 3 for a graphical representation. E &% '$ A &% '$ B FIGURE 3 Quantifiers as Relations A simple binary quantifier D on a domain E is a relation between subsets of E: DE 2 ((E) \Theta (E)) The trivial quantifiers are ?E and ?E , which hold of all and of no pairs of sets, respectively. Not all elements in ((E)\Theta(E)) serve as natural language determiner denotations. In fact, one of the first insights provided by quantification Basic Quantifier Theory / 7 theory is that such determiners hav...
OMITTING UNCOUNTABLE TYPES, AND THE STRENGTH OF [0, 1]VALUED LOGICS
"... Abstract. We study a class of [0, 1]valued logics for continuous metric structures. The main result of the paper is maximality theorem that characterizes these logics in terms of a modeltheoretic property, namely, an extension of the omitting types theorem to uncountable languages. ..."
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Abstract. We study a class of [0, 1]valued logics for continuous metric structures. The main result of the paper is maximality theorem that characterizes these logics in terms of a modeltheoretic property, namely, an extension of the omitting types theorem to uncountable languages.
A Perspective on Lindström Quantifiers and Oracles
"... This paper presents a perspective on the relationship between Lindström quantiers in model theory and oracle computations in complexity theory. We do not study this relationship here in full generality (indeed, there is much more work to do in order to obtain a full appreciation), but instead w ..."
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This paper presents a perspective on the relationship between Lindström quantiers in model theory and oracle computations in complexity theory. We do not study this relationship here in full generality (indeed, there is much more work to do in order to obtain a full appreciation), but instead we examine what amounts to a thread of research in this topic running from the motivating results, concerning logical characterizations of nondeterministic polynomialtime, to the consideration of Lindström quantiers as oracles, and through to the study of some naturally arising questions (and subsequent answers). Our presentation follows the chronological progress of the thread and highlights some important techniques and results at the interface between nite model theory and computational complexity theory.
Coalgebraic Lindström Theorems
"... We study modal Lindström theorems from a coalgebraic perspective. We provide three different Lindström theorems for coalgebraic logic, one of which is a direct generalisation of de Rijke’s result for Kripke models. Both the other two results are based on the properties of bisimulation invariance, ..."
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We study modal Lindström theorems from a coalgebraic perspective. We provide three different Lindström theorems for coalgebraic logic, one of which is a direct generalisation of de Rijke’s result for Kripke models. Both the other two results are based on the properties of bisimulation invariance, compactness, and a third property: ωbisimilarity, and expressive closure at level ω, respectively. These also provide new results in the case of Kripke models. Discussing the relation between our work and a recent result by van Benthem, we give an example showing that only requiring bisimulation invariance together with compactness does not suffice to characterise basic modal logic.