Algebraic Probabilistic Consistency

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.