In recent years, the discovery of the relationships between formulas in Łukasiewicz logic and rational polyhedra, Chang MV-algebras and lattice-ordered abelian roups, MV-algebraic states and coherent de Finetti’s assessments of continuous events, has changed the study and practice of many-valued logic. This book is intended as an up-to-date monograph on infinite-valued Łukasiewicz logic and MV-algebras. Each chapter features a combination of classical and re¬cent results, well beyond the traditional domain of algebraic logic: among others, a comprehensive account is given of many effective procedures that have been re¬cently developed for the algebraic and geometric objects represented by formulas in Łukasiewicz logic. The book embodies the viewpoint that modern Łukasiewicz logic and MV-algebras provide a benchmark for the study of several deep mathematical prob¬lems, such as Rényi conditionals of continuously valued events, the many-valued generalization of Carathéodory algebraic probability theory, morphisms and invari¬ant measures of rational polyhedra, bases and Schauder bases as jointly refinable partitions of unity, and first-order logic with [0,1]-valued identity on Hilbert space. Complete versions are given of a compact body of recent results and techniques, proving virtually everything that is used throughout, so that the book can be used both for individual study and as a source of reference for the more advanced reader.
Author(s): D. Mundici
Series: Trends in Logic 35
Edition: 1
Publisher: Springer Netherlands
Year: 2011
Language: English
Pages: 258
Tags: Mathematical Logic and Foundations
Front Matter....Pages i-xviii
Prologue: de Finetti Coherence Criterion and Łukasiewicz Logic....Pages 1-10
Rational Polyhedra, Interpolation, Amalgamation....Pages 11-25
The Galois Connection (Mod, Th) in Ł $$\infty$$ ....Pages 27-39
The Spectral and the Maximal Spectral Space....Pages 41-53
De Concini–Procesi Theorem and Schauder Bases....Pages 55-68
Bases and Finitely Presented MV-Algebras....Pages 69-79
The Free Product of MV-Algebras....Pages 81-88
Direct Limits, Confluence and Multisets....Pages 89-100
Tensors....Pages 101-118
States and the Kroupa–Panti Theorem....Pages 119-130
The MV-Algebraic Loomis–Sikorski Theorem....Pages 131-140
The MV-Algebraic Stone–von Neumann Theorem....Pages 141-148
Recurrence, Probability, Measure....Pages 149-158
Measuring Polyhedra and Averaging Truth-Values....Pages 159-166
A Rényi Conditional in Łukasiewicz Logic....Pages 167-178
The Lebesgue State and the Completion of $${\mathsf {FREE}}_n$$ ....Pages 179-186
Finitely Generated Projective MV-Algebras....Pages 187-196
Effective Procedures for $$\hbox{\L}_{\infty}$$ and MV-Algebras....Pages 197-213
A First-Order Łukasiewicz Logic with [0, 1]-Identity....Pages 215-225
Applications, Further Reading, Selected Problems....Pages 227-238
Background Results....Pages 239-247
Back Matter....Pages 251-256
