Philosophy and model theory - Button, Tim; Walsh, Sean (Sean P.); Hodges, Wilfrid (UK: Oxford University Press, 2018)
"Model theory is used in every theoretical branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging uses of model theory have created a highly fragmented literature. On the one hand, many philosophically significant results are found only in mathematics textbooks: these are aimed squarely at mathematicians; they typically presuppose that the reader has a serious background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are scattered across disconnected pockets of papers. The first aim of this book, then, is to explore the philosophical uses of model theory, focusing on the central topics of reference, realism, and doxology. Its second aim is to address important questions in the philosophy of model theory, such as: sameness of theories and structure, the boundaries of logic, and the classification of mathematical structures. Philosophy and Model Theory will be accessible to anyone who has completed an introductory logic course. It does not assume that readers have encountered model theory before, but starts right at the beginning, discussing philosophical issues that arise even with conceptually basic model theory. Moreover, the book is largely self-contained: model-theoretic notions are defined as and when they are needed for the philosophical discussion, and many of the most philosophically significant results are given accessible proofs."-- Preface, page [v].
A. Reference and realism. Logics and languages -- Permutations and referential indeterminacy -- Ramsey sentences and Newman's objection -- Compactness, infinitesimals, and the reals -- Sameness of structure and theory -- B. Categoricity. Modelism and mathematical doxology -- Categoricity and the natural numbers -- Categoricity and the sets -- Transcendental arguments against model-theoretical scepticism -- Internal categoricity and the natural numbers -- Internal categoricity and the sets -- Internal categoricity and truth -- Boolean-valued structures -- C. Indiscernibility and classification. Types and Stone spaces -- Indiscernibility -- Quantifiers -- Classification and uncountable categoricity -- D. Historical appendix. Wilfrid Hodges.
Includes bibliographical references (pages 477-505) and indexes.
Keywords: Logic, Symbolic and mathematical; Model theory