This book investigates the foundations of probability theory and logic, intertwining historical insights with modern interpretations. It explores the evolution of probability theory from Boole’s seminal question on the very object of probability, through de Finetti’s finitely additive probability and his consistency notion, also known as “non-Dutchbookability,” to the intricate relationship between logic independence and stochastic independence. Using the recent characterization of Lukasiewicz logic as the only logic generated by a continuous [ 0,1]-valued operation having the two minimal properties of what is commonly understood as an implication, the author extends the results of the first part of the book from yes-no events to continuous real-valued events. The book culminates with a detailed examination of the symbiosis between de Finetti’s finitely additive and Kolmogorov’s countably additive probability on compact spaces. By providing a rigorous and cohesive narrative, this book serves as an essential resource for scholars and students in mathematical logic eager to grasp the profound connections between logic, probability, and algebraic structures.
Geometry of finite boolean algebras and their states.- De Finettis
Fundamental Theorem of Probability.- De Finettis Consistency Theorem.-
Boolean independence, consistency, and the product law.- Interlude: de
Finettis exchangeability theorem.- The logic L of continuous [ 0, 1] valued
events.- MV algebraic probabilistic consistency.- The product law for
continuous [ 0, 1] events.- Finite/countable additivity.
Daniele Mundici is an academic researcher from the University of Florence. He has contributed to research in ukasiewicz logic, Chang MV-algebras, lattice-ordered groups, AF C*-algebras and their computational complexity. The author has an h-index of 28, and co-authored 203 publications. Previous affiliations of Daniele Mundici include the Department of Computer Science of the University of Milan. He has served as a president of the Kurt Gödel Society.
