zbMATH — the first resource for mathematics

A logical characterization of coherence for imprecise probabilities. (English) Zbl 1244.03082
This 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 \([0,1]\subset\mathbb R.\)

03B48 Probability and inductive logic
03B50 Many-valued logic
06D35 MV-algebras
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.