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Mathematics | Advanced Łukasiewicz calculus and MV-algebras

Advanced Łukasiewicz calculus and MV-algebras

Series: Trends in Logic, Vol. 35

University of Florence, Prof. D. Mundici

2011, XVIII, 258 p.

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  • Written for self-study
  • References the self-contained book Trends in Logic 7 (co-authored by the same author)
  • Deals with the logic and probability of continuously-valued events, just as boolean logic deals with yes-no events
  • Focuses on Lukasiewicz logic and their algebras, because of the powerful mathematical tools available here
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.

Content Level » Research

Keywords » MV-algebras - conditioning - fuzzy logic - many-valued logic - probability of continuously valued events

Related subjects » Mathematics

Table of contents 

Preface.- Chapter 1. Prologue: de Finetti coherence criterion and Łukasiewicz logic.- Chapter 2. Rational polyhedra, Interpolation, Amalgamation.- Chapter 3. The Galois connection (Mod, Th) in Ł∞ 21.- Chapter 4. The spectral and the maximal spectral space.- Chapter 5. De Concini-Procesi theorem and Schauder bases.- Chapter 6. Bases and finitely presented MV-algebras.- Chapter 7. The free product of MV-algebras.- The construction of free products.- Chapter 8. Direct limits, confluence and multisets.-  Chapter 9. Tensors.- Chapter 10. States and the Kroupa-Panti Theorem.- Chapter 11. The MV-algebraic Loomis-Sikorski theorem.-  Chapter 12. The MV-algebraic Stone-von Neumann theorem.- Chapter 13. Recurrence, probability, measure.- Chapter 14. Measuring polyhedra and averaging truth-values.- Chapter 15. A Rényi conditional in Łukasiewicz logic.- Chapter 16. The Lebesgue state and the completion of FREEn.- Chapter 17. Finitely generated projective MV-algebras.- Chapter 18. Effective procedures for Ł∞ and MV-algebras.- Chapter 19. A first-order Łukasiewicz logic with [0, 1]-identity.- Chapter 20. Applications, further reading, selected problems.- Chapter 21. Background results.- Special Bibliography. References. Index.

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