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Generalized Probabilistic Modus Ponens

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Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10369))

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Abstract

Modus ponens (from A and “if A then C” infer C) is one of the most basic inference rules. The probabilistic modus ponens allows for managing uncertainty by transmitting assigned uncertainties from the premises to the conclusion (i.e., from P(A) and P(C|A) infer P(C)). In this paper, we generalize the probabilistic modus ponens by replacing A by the conditional event A|H. The resulting inference rule involves iterated conditionals (formalized by conditional random quantities) and propagates previsions from the premises to the conclusion. Interestingly, the propagation rules for the lower and the upper bounds on the conclusion of the generalized probabilistic modus ponens coincide with the respective bounds on the conclusion for the (non-nested) probabilistic modus ponens.

G. Sanfilippo—Partially supported by INdAM–GNAMPA Project 2016 Grant U 2016/000391

N. Pfeifer—Supported by his DFG project PF 740/2-2 (within the SPP1516)

A. Gilio—Retired

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Acknowledgments

We thank three anonymous referees for their useful comments and suggestions. We thank DFG, FMSH, and Villa Vigoni for supporting joint meetings at Villa Vigoni where parts of this work originated (Project: “Human Rationality: Probabilistic Points of View”).

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Palermo, Palermo, Italy

    Giuseppe Sanfilippo

  2. Munich Center for Mathematical Philosophy, LMU Munich, Munich, Germany

    Niki Pfeifer

  3. Department SBAI, University of Rome “La Sapienza”, Rome, Italy

    Angelo Gilio

Authors
  1. Giuseppe Sanfilippo
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  2. Niki Pfeifer
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  3. Angelo Gilio
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Correspondence to Giuseppe Sanfilippo .

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Editors and Affiliations

  1. IDSIA, Lugano, Switzerland

    Alessandro Antonucci

  2. ONERA, Toulouse, France

    Laurence Cholvy

  3. Aix-Marseille University, Marseille, France

    Odile Papini

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Sanfilippo, G., Pfeifer, N., Gilio, A. (2017). Generalized Probabilistic Modus Ponens. In: Antonucci, A., Cholvy, L., Papini, O. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2017. Lecture Notes in Computer Science(), vol 10369. Springer, Cham. https://doi.org/10.1007/978-3-319-61581-3_43

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  • DOI: https://doi.org/10.1007/978-3-319-61581-3_43

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