A logico-geometrical approach to (subjective) uncertainty

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Bio. Tommaso Flaminio is a Ramón y Cajal Researcher at the Artificial Research Institute (IIIA) of the Spanish National Research Council (CSIC), Barcelona, Spain. He obtained his Laurea degree in Mathematics at the University of Siena (Italy) and the PhD in Mathematical Logic and Theoretical Computer Science in the same university. Before the Ramón y Cajal, Tommaso Flaminio spent three more postdoc periods: one at the University of Insubria (Varese, Italy) within the project "Probability Theory of Nonclassical Events" led by Vincenzo Marra; a second one in Barcelona (Spain, 2010-2013) under the supervision of Lluis Godo at the IIIA-CSIC institute for Artificial Intelligence with a Juan de la Cierva research project titled "Reasoning under uncertainty: a measure theoretical, logical and algebraic approach", and third one in Siena (Italy, 2007-2009) under the supervision of Franco Montagna at the Department of Mathematics and Computer Science. Since January 2017 Tommaso Flaminio is coordinator of the Working Group on Mathematical Fuzzy Logic of the European Society for Fuzzy Logic and Technology EUSFLAT. His research interests are in logic, especially algebraic and modal logics, and subjective probability on which he grounded his main research direction on problems arising from uncertain reasoning, artificial intelligence and decision theory.

Resumen. De Finetti defines the probability of an unknown event as the fair price which a rational gambler is willing to pay to participate in a betting game against the bookmaker, the payoffs of which are 1 in case the event occurs, and 0 otherwise. Based on this very simple idea, de Finetti showed that all theorems of probability theory may be derived as consequences of his coherence condition on probability assignments (regarded as prices) on logically connected events. In this talk I push forward a general geometric approach to coherence and, by doing so, I provide operational foundations for uncertainty measures focusing, in particular, on states, necessity and possibility measures, and Dempster-Shafer's belief functions on MV-algebras of real-valued events. These results highlight the geometric significance of Eucliedean geometry for probability theory and pave the way for a series of generalizations. Indeed, modifying the geometrical background from Euclidean to tropical geometry, a general theorem a la de Finetti can be obtained for several classes of uncertainty measures on real-valued events.