Final Report Summary - RETHINKING (Rethinking uncertainty: A problem-based approach) The RETHINKING project has been carried out according to the proposed schedule and produced satisfactory results, both in terms of research output (three peer-reviewed journal publications, two peer-reviewed conference proceedings and one peer-reviewed book chapter) and in terms of the fellow's own career development. At the end of the project the fellow accepted a permanent associate professorship in Logic and Philosophy of Science at the University of Milan.1.1 ResultsThe project included three research objectives. Objective 1 (Foundations) covered much of the ground work for the results obtained in the project. It also led to results of independent conceptual interest, and in particular to a defence of Bayesian approach to second-orded uncertainty. Besides the planned work-package an unexpected line of research was developed in relation to this Objective. This involves providing a choice-based semantics to real-valued logics and investigating its relation with the notion of objective probability. This has been the topic of a new collaboration set up between the fellow and Prof. Vincenzo Marra (Milan), along with a jointly supervised PhD student, Rossella Marrano (Pisa).In addition, Objective 1 allowed the fellow to relate the problem of rethinking uncertainty with the use of mathematical models in the social sciences, and in particular to economic theory. This has been the topic of a collaboration with Prof. Mariano Giaquinta (Pisa). As a final, unexpected, result in the foundational part of the project, the author (in collaboration with T. Flaminio and Ll. Godo) investigated a novel algebraic approach to the definition of conditional events. This will constitute the basis for future work on conditional choice-based measures of uncertainty. Objective 2 (Choice-problems) aimed at fleshing out the choice-based roots of the standard Bayesian approach to the quantification of uncertainty. This work-package has been carried out building on an existing collaboration between the fellow and Dr. Tommaso Flaminio (Univerisity of Insubria) and Prof. Lluis Godo (IIIA, Barcelona). The key result include the definition and characterisation of "Choice-based probabilities" and a detailed analysis of the way in which they refine standard probability functions. Objective 3 (Extensions) aimed at investigating how the choice-based foundation of uncertainty measures should lead to extending the classical notion of probability. In response to this, the fellow (again in collaboration with T. Flaminio and Ll. Godo) introduced a general framework for defining non-probabilistic measures of uncertainty on the basis of suitably specified betting problems. In particular this led to an essentially "Bayesian" foundation to non-additive measures (a.k.a Belief Functions). Another result related to the extension of the classical Bayesian framework was obtained by the fellow in collaboration with Prof. Franco Montagna (Siena). It consists in the definition of a betting criterion leading to the characterisation of imprecise, non-standard probabilities on many-valued events. Finally, the use of generalised measures of uncertainty in the characterisation of rational decision-making, has been the object of a new collaboration with Dr. Enrico Marchioni (University of Toulouse). The key results in this line of research involve a novel approach to the selection of multiple Nash equilibria based on possibility measures.
