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Algebraic Probabilistic Consistency: Boole, Lukasiewicz, de Finetti, Kolmogorov

By (author) Daniele Mundici

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This book investigates the foundations of probability theory and logic, intertwining historical insights with modern interpretations. It explores the evolution of probability theory from Booleâ€™s seminal question on the very object of probability, through de Finettiâ€™s finitely additive probability and his consistency notion, also known Â as Â â€œnon-Dutchbookability,â€ to the intricate relationship between logic independence and stochastic independence. Â Using the recent characterization of Åukasiewicz 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 Finettiâ€™s finitely additive and Kolmogorovâ€™s countably additive probability on compact spaces. Â 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.

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