Philosophy and model theory /
Model theory is an important area of mathematical logic which has deep philosophical roots, many philosophical applications, and great philosophical interest in itself. The aim of this book is to introduce, organise, survey, and develop these connections between philosophy and model theory, for the...
| Main Authors: | , |
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| Format: | eBook |
| Language: | English |
| Published: |
Oxford, United Kingdom :
Oxford University Press,
2018.
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| Edition: | First edition. |
| Subjects: | |
| Online Access: | Connect to the full text of this electronic book |
Table of Contents:
- Cover; Philosophy and Model Theory; Copyright; Preface; Topic selection; Structuring the book; Presuppositions and proofs; Acknowledgements; Contents; A: Reference and realism; Introduction to Part A; 1: Logics and languages; 1.1 Signatures and structures; 1.2 First-order logic: a first look; Syntax for first-order logic; Semantics: the trouble with quantifiers; Why it is worth considering different approaches; 1.3 The Tarskian approach to semantics; 1.4 Semantics for variables; 1.5 The Robinsonian approach to semantics; 1.6 Straining the notion of â#x80;#x98;languageâ#x80;#x99
- 1.7 The Hybrid approach to semantics1.8 Linguistic compositionality; 1.9 Second-order logic: syntax; 1.10 Full semantics; 1.11 Henkin semantics; 1.12 Consequence; 1.13 Definability; 1.A First- and second-order arithmetic; 1.B First- and second-order set theory; 1.C Deductive systems; 2: Permutations and referential indeterminacy; 2.1 Isomorphism and the Push-Through Construction; 2.2 Benacerraf â#x80;#x99;s use of Push-Through; 2.3 Putnamâ#x80;#x99;s use of Push-Through; The permutation argument; Preferable models; Referential indeterminacy for moderate objects-platonism
- 2.4 Attempts to secure reference in mathematicsShapiroâ#x80;#x99;s ante rem structuralism; Putnamâ#x80;#x99;s internal realism (constructivist reading); Syntactic Priority; 2.5 Supervaluationism and indeterminacy; 2.6 Conclusion; 2.A Eligibility, definitions, and Completeness; 2.B Isomorphism and satisfaction; 3: Ramsey sentences and Newmanâ#x80;#x99;s objection; 3.1 The o/t dichotomy; 3.2 Ramsey sentences; 3.3 The promise of Ramsey sentences; 3.4 A caveat on the o/t dichotomy; 3.5 Newmanâ#x80;#x99;s criticism of Russell; 3.6 The Newman-conservation-objection; 3.7 Observation vocabulary versus observable objects
- 3.8 The Newman-cardinality-objection3.9 Mixed-predicates again: the case of causation; 3.10 Natural properties and just more theory; 3.A Newman and elementary extensions; 3.B Conservation in first-order theories; 4: Compactness, infinitesimals, and the reals; 4.1 The Compactness Theorem; 4.2 Infinitesimals; 4.3 Notational conventions; 4.4 Differentials, derivatives, and the use of infinitesimals; 4.5 The orders of infinite smallness; 4.6 Non-standard analysis with a valuation; 4.7 Instrumentalism and conservation; 4.8 Historical fidelity; 4.9 Axiomatising non-standard analysis
- Axiomatising elementary extensionsAxiomatising the reals; Axiomatising non-standardness; 4.10 Axiomatising the reals; 4.A Gödelâ#x80;#x99;s Completeness Theorem; 4.B A model-theoretic proof of Compactness; 4.C The valuation function of Â4.6; 5: Sameness of structure and theory; 5.1 Definitional equivalence; 5.2 Sameness of structure and ante rem structuralism; 5.3 Interpretability; 5.4 Biinterpretability; 5.5 From structures to theories; 5.6 Interpretability and the transfer of truth; Counterexamples to the Truth-Transfer Thesis; Bridge principles; Difficulties concerning bridge principles