Conditionals and Modularity in General Logics [electronic resource] /
Gabbay, Dov M.author.
Schlechta, Karl.author.
This text centers around three main subjects. The first is the concept of modularity and independence in classical logic and nonmonotonic and other nonclassical logic, and the consequences on syntactic and semantical interpolation and language change. In particular, we will show the connection between interpolation for nonmonotonic logic and manipulation of an abstract notion of size. Modularity is essentially the ability to put partial results achieved independently together for a global result. The second aspect of the book is the authors' uniform picture of conditionals, including many-valued logics and structures on the language elements themselves and on the truth value set. The third topic explained by the authors is neighbourhood semantics, their connection to independence, and their common points and differences for various logics, e.g., for defaults and deontic logic, for the limit version of preferential logics, and for general approximation. The book will be of value to researchers and graduate students in logic and theoretical computer science.
Description based upon print version of record.
Conditionals and Modularityin General Logics; Contents; Chapter 1: Introduction; 1.1 The Main Subjects of This Book; 1.1.1 An Example; 1.1.1.1 An Abstract Description of Both Cases in the Above Example; 1.1.2 Connections; 1.2 Main Definitions and Results; 1.2.1 The Monotonic Case; 1.2.2 The Non-monotonic Case; 1.3 Overview of This Introduction; 1.4 Basic Definitions; 1.5 Towards a Uniform Picture of Conditionals; 1.5.1 Discussion and Classification; 1.5.2 Additional Structure on Language and Truth Values; 1.5.3 Representation for General Revision, Update, and Counterfactuals
1.6 Interpolation1.6.1 Introduction; 1.6.1.1 Background; 1.6.2 Problem and Method; 1.6.3 Monotone and Antitone Semantic and Syntactic Interpolation; 1.6.3.1 Semantic Interpolation; 1.6.3.2 The Interval of Interpolants; 1.6.3.3 Syntactic Interpolation; 1.6.3.4 Finite Goedel Logics; 1.6.4 Laws About Size and Interpolation in Non-monotonic Logics; 1.6.4.1 Various Concepts of Size and Non-monotonic Logics; 1.6.4.2 Additive and Multiplicative Laws About Size; 1.6.4.3 Interpolation and Size; 1.6.4.4 Hamming Relations and Size; 1.6.4.5 Equilibrium Logic
1.6.4.6 Interpolation for Revision and Argumentation1.6.4.7 Language Change to Obtain Products; 1.6.5 Summary; 1.7 Neighbourhood Semantics; 1.7.1 Defining Neighbourhoods; 1.7.2 Additional Requirements; 1.7.3 Connections Between the Various Properties; 1.7.4 Various Uses of Neighbourhood Semantics; 1.8 An Abstract View on Modularity and Independence; 1.8.1 Introduction; 1.8.2 Abstract Definition of Independence; 1.8.2.1 Discussion; 1.8.3 Other Aspects of Independence; 1.8.3.1 Existence of Normal Forms; 1.8.3.2 Language Change; 1.8.3.3 A Relevance Problem; 1.8.3.4 Small Subspaces
1.9 Conclusion and Outlook1.10 Previously Published Material, Acknowledgements; Chapter 2: Basic Definitions; 2.1 Introduction; 2.1.1 Overview of This Chapter; 2.2 Basic Algebraic and Logical Definitions; 2.2.1 Countably Many Disjoint Sets; 2.2.2 Introduction to Many-Valued Logics; 2.2.2.1 Definable Model Functions; 2.2.2.2 Generalization of Model Sets and (In)essential Variables, Overview; 2.2.2.3 Interpolation of Many-Valued Logics; 2.3 Preferential Structures; 2.3.1 The Minimal Variant; 2.3.1.1 New Material on the Minimal Variant of Preferential Structures (Bubble Structures)
2.3.2 The Limit Variant2.3.2.1 New Material on the Limit Variant of Preferential Structures; 2.3.3 Preferential Structures for Many-Valued Logics; 2.4 IBRS and Higher Preferential Structures; 2.4.1 General IBRS; 2.4.2 Higher Preferential Structures; 2.4.2.1 Introduction; 2.4.2.2 Definitions and Facts for Basic Structures; 2.4.2.3 Definitions and Facts for Totally and Essentially Smooth Structures; 2.4.2.4 Semantic Representation Results for Higher Preferential Structures; 2.4.2.5 Translation to Logics; 2.4.2.6 Discussion of the Limit Version of Higher Preferential Structures
2.4.2.7 Ideas for the Intuitive Meaning of Higher Preferential Structures
Includes bibliographical references and index.
This text centers around three main subjects. The first is the concept of modularity and independence in classical logic and nonmonotonic and other nonclassical logic, and the consequences on syntactic and semantical interpolation and language change. In particular, we will show the connection between interpolation for nonmonotonic logic and manipulation of an abstract notion of size. Modularity is essentially the ability to put partial results achieved independently together for a global result. The second aspect of the book is the authors' uniform picture of conditionals, including many-valued logics and structures on the language elements themselves and on the truth value set. The third topic explained by the authors is neighbourhood semantics, their connection to independence, and their common points and differences for various logics, e.g., for defaults and deontic logic, for the limit version of preferential logics, and for general approximation. The book will be of value to researchers and graduate students in logic and theoretical computer science.
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