Quantifiers : logics, models, and computation

Quantifiers: Logics, Models and Computation is the first concentrated effort to give a systematic presentation of the main research results on the subject, since the modern concept was formulated in the late '50s and early '60s. The majority of the papers are in the nature of a handbook. All of them are self-contained, at various levels of difficulty. The Introduction surveys the main ideas and problems encountered in the logical investigation of quantifiers. The Prologue, written by Per Lindstroem, presents the early history of the concept of generalised quantifiers. The volume then continues with a series of papers surveying various research areas, particularly those that are of current interest. Together they provide introductions to the subject from the points of view of mathematics, linguistics, and theoretical computer science. The present volume has been prepared in parallel with Quantifiers: Logics, Models and Computation, Volume Two. Contributions, which contains a collection of research papers on the subject in areas that are too fresh to be summarised. The two volumes are complementary. For logicians, mathematicians, philosophers, linguists and computer scientists. Suitable as a text for advanced undergraduate and graduate specialised courses in logic.

• Preface. Quantifiers, Some Problems and Ideas
• M. Krynicki, M. Mostowski. Prologue
• P. Lindstroem. On the Model Theory of Some Generalized Quantifiers
• H.D. Ebbinghaus. Finite Generation Problem and n-ary Quantifiers
• L. Hella, K. Luosto. Games and Trees in Infinitary Logic: a Survey
• J. Vaanaen. Theory of Linear Order in Extended Logics
• H. Herre. Henkin Quantifiers
• M. Krynicki, M. Mostowski. Continuous Operations on Spaces of Structures
• X. Caicedo. Model Theory of Topological Structures
• J. Flum. Computable Quantifiers and Logics over Finite Structures
• J.A. Makowski, Y.B. Pnueli. Quantifiers in Natural Language. A Survey of Some Recent Work
• D. Westerstahl. Index of Names.

This volume contains a collection of research papers centered around the concept of quantifier. Recently this concept has become the central point of research in logic. It is one of the important logical concepts whose exact domain and applications have so far been insufficiently explored, especially in the area of inferential and semantic properties of languages. It should thus remain the central point of research in the future. Moreover, during the last twenty years generalized quantifiers and logical technics based on them have proved their utility in various applications. The example of natu rallanguage semantics has been partcularly striking. For a long time it has been belived that elementary logic also called first-order logic was an ade quate theory of logical forms of natural language sentences. Recently it has been accepted that semantics of many natural language constructions can not be properly represented in elementary logic. It has turned out, however, that they can be described by means of generalized quantifiers. As far as computational applications oflogic are concerned, particulary interesting are semantics restricted to finite models. Under this restriction elementary logic looses several of its advantages such as axiomatizability and compactness. And for various purposes we can use equally well some semantically richer languages of which generalized quantifiers offer the most universal methods of describing extensions of elementary logic. Moreover we can look at generalized quantifiers as an explication of some specific mathematical concepts, e. g.

• Preface. Quantifiers and Inference
• J. van Benthem. Operators on Branched Quantifiers
• M.A. Brown. Hilbert's Epsilon-Symbol in the Presence of Generalized Quantifiers
• X. Caicedo. Partially Ordered Connectives and Finite Graphs
• L. Hella, G. Sandu. Theories of Finitely Determined Linear Orderings in Stationary Logics
• H. Herre. Definable Second-Order Quantifiers and Quasivarieties
• A.A. Ivanov. Quantifiers Determined by Classes of Binary Relations
• M. Krynicki. Decidability Results for Classes of Ordered Abelian Groups in Logics with Ramsey Quantifiers
• W. Lenski. On the Eliminability of the Quantifier `There Exists Uncountably Many'
• Z. Mijajlovic. Quantifiers Definable by Second Order Means
• M. Mostowski. Generalized Quantifiers in Algebra
• A.G. Pinus. On Ordering of the Family of Logics with Skolem-Loewenheim Property and Countable Compactness Property
• M. Waclawek.DAGGER. Pre-Ordered Quantifiers in Elementary Sentences of Natural Language
• M.W. Zawadowsky. Some Remarks on Zawadowski's `Theory of Preordered Quantifiers'
• M.A. Brown. Index of Names.

The quantifier concept is central to current logical investigations. Quantifiers: Logics, Models and Computation. Volume Two: Contributions contains twelve research papers devoted to generalized quantifiers and their applications. It offers an exhaustive survey of logical methods that are useful in investigations involving generalized quantifiers. Both model-theoretical and proof-theoretical approaches are well represented. Moreover, some papers focus on the applications of logical theory, particularly in relation to semantics of natural language. The volume includes a wide-ranging survey of logical methods which are useful in investigations into generalized quantifiers. The book is the second volume of a large collection. The first volume - Quantifiers: Logics, Models and Computation. Volume One: Surveys - contains a collection of survey papers on selected, well-defined areas organized around the quantifier concept. Volumes One and Two are complementary. For logicians, mathematicians, philosophers, linguists and computer scientists. Also suitable as a text for advanced undergraduate or graduate specialized courses in logic.

• Preface. Quantifiers and Inference
• J. van Benthem. Operators on Branched Quantifiers
• M.A. Brown. Hilbert's Epsilon-Symbol in the Presence of Generalized Quantifiers
• X. Caicedo. Partially Ordered Connectives and Finite Graphs
• L. Hella, G. Sandu. Theories of Finitely Determined Linear Orderings in Stationary Logics
• H. Herre. Definable Second-Order Quantifiers and Quasivarieties
• A.A. Ivanov. Quantifiers Determined by Classes of Binary Relations
• M. Krynicki. Decidability Results for Classes of Ordered Abelian Groups in Logics with Ramsey Quantifiers
• W. Lenski. On the Eliminability of the Quantifier `There Exists Uncountably Many'
• Z. Mijajlovic. Quantifiers Definable by Second Order Means
• M. Mostowski. Generalized Quantifiers in Algebra
• A.G. Pinus. On Ordering of the Family of Logics with Skolem-Loewenheim Property and Countable Compactness Property
• M. Waclawek.DAGGER. Pre-Ordered Quantifiers in Elementary Sentences of Natural Language
• M.W. Zawadowsky. Some Remarks on Zawadowski's `Theory of Preordered Quantifiers'
• M.A. Brown. Index of Names.