A logical characterization of coherence for imprecise probabilities.

*(English)*Zbl 1244.03082This dense paper provides a logico-algebraic framework which allows for a logical characterization of coherence for imprecise improbability.

The first chapter begins with a discussion of first-order and second-order uncertainty. It is argued that in many interesting situations single-valued probabilities and/or binary events cannot satisfy practical needs. To combine statistical and logical approaches to second-order uncertainty the authors aim to develop a framework allowing to represent key concepts of both traditions.

After a discussion of de Finetti’s Dutch Book argument, the authors give brief introductions to the logical tradition using probability over many-valued events and MV-algebras and the statistical tradition using previsions.

The first part of the second chapter contains a highly condensed presentation of algebraic tools such as MV-algebra topology and filters on MV-algebras. In the second part a logical characterization of coherence of assessments of upper/lower probability and previsions are given.

In Chapter 3 external operators of upper and lower probability are internalized in the algebra yielding a UMV-algebra. These UMV-algebras are then used to characterize coherence.

Chapter 4 begins with the definition of UG-algebras, which are obtained from UMV-algebras internalizing real-valued upper previsions. It is then shown that there exists a categorical equivalence between UG-algebras and UMV-algebras. This equivalence is then exploited to prove the existence of PTIME-computable maps \(\circ\) and \(*\) between UG-equations and UMV-equations which preserve semantic consequence and satisfiability.

Chapter 5 contains various complexity results concerning UMV-algebras. For instance, it is proved that satisfiability of certain UMV-equations is NP-complete.

In Chapter 6 the authors conclude and point out further promising lines of research, such as a logico-algebraic framework with conditioning events taking values in \([0,1]\subset\mathbb R.\)

The first chapter begins with a discussion of first-order and second-order uncertainty. It is argued that in many interesting situations single-valued probabilities and/or binary events cannot satisfy practical needs. To combine statistical and logical approaches to second-order uncertainty the authors aim to develop a framework allowing to represent key concepts of both traditions.

After a discussion of de Finetti’s Dutch Book argument, the authors give brief introductions to the logical tradition using probability over many-valued events and MV-algebras and the statistical tradition using previsions.

The first part of the second chapter contains a highly condensed presentation of algebraic tools such as MV-algebra topology and filters on MV-algebras. In the second part a logical characterization of coherence of assessments of upper/lower probability and previsions are given.

In Chapter 3 external operators of upper and lower probability are internalized in the algebra yielding a UMV-algebra. These UMV-algebras are then used to characterize coherence.

Chapter 4 begins with the definition of UG-algebras, which are obtained from UMV-algebras internalizing real-valued upper previsions. It is then shown that there exists a categorical equivalence between UG-algebras and UMV-algebras. This equivalence is then exploited to prove the existence of PTIME-computable maps \(\circ\) and \(*\) between UG-equations and UMV-equations which preserve semantic consequence and satisfiability.

Chapter 5 contains various complexity results concerning UMV-algebras. For instance, it is proved that satisfiability of certain UMV-equations is NP-complete.

In Chapter 6 the authors conclude and point out further promising lines of research, such as a logico-algebraic framework with conditioning events taking values in \([0,1]\subset\mathbb R.\)

Reviewer: Jürgen Landes (Canterbury)

##### MSC:

03B48 | Probability and inductive logic |

03B50 | Many-valued logic |

06D35 | MV-algebras |

68Q17 | Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) |

##### Keywords:

subjective probability; probabilistic logic; coherence; imprecise probability; MV-algebra; betting framework; many-valued logic; second-order uncertainty; satisfiability
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\textit{M. Fedel} et al., Int. J. Approx. Reasoning 52, No. 8, 1147--1170 (2011; Zbl 1244.03082)

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##### References:

[1] | Aguzzoli, S.; Gerla, B.; Marra, V., De finetti’s no-Dutch-book criterion for Gödel logic, Studia logica, 90, 25-41, (2008) · Zbl 1165.03008 |

[2] | Blok, W.; Pigozzi, D., Algebraizable logics, Memoirs of the American mathematical society, 396, 77, (1989) · Zbl 0664.03042 |

[3] | Burris, S.; Sankappanavar, H.P., A course in universal algebra, (1981), Springer Verlag New York · Zbl 0478.08001 |

[4] | Chang, C.C., A new proof of the completeness of łukasiewicz axioms, Transactions of the American mathematical society, 93, 74-80, (1989) · Zbl 0093.01104 |

[5] | Cignoli, R.; D’Ottaviano, I.; Mundici, D., Algebraic foundations of many-valued reasoning, (2000), Kluwer Dordrecht · Zbl 0937.06009 |

[6] | de Cooman, G., Belief models: an order-theoretic investigation, Annals of mathematics and artificial intelligence, 45, 5-34, (2005) · Zbl 1094.03007 |

[7] | de Cooman, G.; Hermans, F., Imprecise probability trees: bridging two theories of imprecise probability, Artificial intelligence, 172, 1400-1427, (2008) · Zbl 1183.60003 |

[8] | de Finetti, B., Theory of probability, vols. I-II, (1974), John Wiley and Sons Chichester |

[9] | de Finetti, B., Sul significato soggettivo Della probabilità, Fundamenta mathematicae, 17, 289-329, (1931) · JFM 57.0608.07 |

[10] | Fedel, M.; Flaminio, T., Non-reversible betting games on fuzzy events: complexity and algebra, Fuzzy sets and systems, 169, 91-104, (2011) · Zbl 1237.91064 |

[11] | M. Fedel, K. Keimel, F. Montagna, W.D. Roth, Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic, Forum Mathematicum. Available from: <http://dx.doi.org/10.1515/FORM.2011.123>. · Zbl 1266.06011 |

[12] | Fine, T.L., Theories of probability, (1973), Academic Press New York |

[13] | Flaminio, T.; Montagna, F., MV-algebras with internal states and probabilistic fuzzy logics, International journal of approximate reasoning, 50, 138-152, (2009) · Zbl 1185.06007 |

[14] | Gerla, B., MV-algebras, multiple bets and subjective states, International journal of approximate reasoning, 1, 1-13, (2000) · Zbl 0958.06007 |

[15] | Gillies, D., Philosophical theories of probability, (2000), Routledge |

[16] | Gillett, P.; Scherl, R.; Shafer, G., A probabilistic logic based on the acceptability of gamble, International journal of approximate reasoning, 44, 3, 281-300, (2007) · Zbl 1115.68148 |

[17] | Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Dordrecht · Zbl 0937.03030 |

[18] | Halpern, J.Y., Reasoning about uncertainty, (2003), MIT Press · Zbl 1090.68105 |

[19] | Howson, C.; de Finetti, B., Countable additivity, consistency and coherence, British journal for the philosophy of science, 59, 1-23, (2008) · Zbl 1142.03309 |

[20] | Howson, C., Can logic be combined with probability? probably, Journal of applied logic, 7, 2, 177-187, (2009) · Zbl 1171.03315 |

[21] | Kroupa, T., Every state on a semisimple MV algebra is integral, Fuzzy sets and systems, 157, 20, 2771-2782, (2006) · Zbl 1107.06007 |

[22] | Kühr, J.; Mundici, D., De Finetti theorem and Borel states in [0,1]-valued algebraic logic, International journal of approximate reasoning, 46, 3, 605-616, (2007) · Zbl 1189.03076 |

[23] | Kumar, A.; Fine, T.L., Stationary lower probabilities and unstable averages, Probability theory and related fields, 69, 1, 1-17, (1985) · Zbl 0535.60006 |

[24] | Levi, I., Imprecision and indeterminacy in probability judgment, Philosophy of science, 52, 3, 390-409, (1985) |

[25] | Marra, V., Is there a probability theory of many-valued events?, () · Zbl 1206.03024 |

[26] | Miranda, E., Updating coherent previsions on finite spaces, Fuzzy sets and systems, 160, 9, 1286-1307, (2009) · Zbl 1180.28009 |

[27] | Mundici, D., Averaging the truth value in łukasiewicz logic, Studia logica, 55, 1, 113-127, (1995) · Zbl 0836.03016 |

[28] | Mundici, D., Tensor product and the loomis sikorski theorem for MV-algebras, Advances in applied mathematics, 22, 227-248, (1999) · Zbl 0926.06004 |

[29] | Mundici, D., Bookmaking over infinite-valued events, International journal of approximate reasoning, 46, 223-240, (2006) · Zbl 1123.03011 |

[30] | Panti, G., Invariant measures in free MV algebras, Communications in algebra, 36, 2849-2861, (2008) · Zbl 1154.06008 |

[31] | Paris, J.B., The uncertain reasoner’s companion: a mathematical perspective, (1994), Cambridge University Press · Zbl 0838.68104 |

[32] | J.B. Paris, A note on the dutch book method, in: T.S. Gert De Cooman, Terrence Fine (Eds.), ISIPTA ’01, Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications, Ithaca, NY, USA, Shaker, 2001. |

[33] | Ramsey, F.P., Truth and probability, () · Zbl 1152.01305 |

[34] | Rudin, W., Functional analysis, (1991), McGraw-Hill Publ. · Zbl 0867.46001 |

[35] | Savage, L., The foundations of statistics, (1954), Chapman and Hall London · Zbl 0055.12604 |

[36] | Shafer, G.; Vovk, V., Probability and finance its only a game!, (2001), John Wiley and Sons New York · Zbl 0985.91024 |

[37] | Shafer, G.; Gillett, P.; Scherl, R.B., A new understanding of subjective probability and its generalization to lower and upper prevision, International journal of approximate reasoning, 33, 1, 1-49, (2003) · Zbl 1092.68098 |

[38] | Smith, C., Consistency in statistical inference and decision, Journal of the royal statistical society. series B (methodological), 23, 1, 1-37, (1961) · Zbl 0124.09603 |

[39] | Walley, P., Statistical reasoning with imprecise probabilities, Monographs on statistics and applied probability, vol. 42, (1991), Chapman and Hall London · Zbl 0732.62004 |

[40] | Walley, P.; Fine, T., Varieties of modal (classificatory) and comparative probability, Synthese, 41, 3, 321-374, (1979) · Zbl 0442.60005 |

[41] | Williams, P.M., Notes on conditional prevision, International journal of approximate reasoning, 44, 3, 363-383, (2007) · Zbl 1114.60005 |

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