Partially undetermined many-valued events and their conditional probability.

*(English)*Zbl 1261.03096In [D. Dubois and H. Prade, “Conditional objects as nonmonotonic consequence relations”, IEEE Trans. Syst. Man Cybern. 24, No. 12, 1724–1740 (1994)], logic for classical conditional events was investigated. In their approach, the truth value of a conditional event may be undetermined. Now this theory is extended to many-valued events; both events in a conditional statement can be many-valued and can be undetermined. Even the value of undeterminacy can be expressed in terms of many-valued logic. In contrast to D. Mundici’s approach [Trends Log. Stud. Log. Libr. 28, 213–232 (2009; Zbl 1163.03016)], the approach presented here leads to an algebra of conditionals. Compared to [G. Coletti and R. Scozzafava, Probabilistic logic in a coherent setting. Dordrecht: Kluwer Academic Publishers (2003; Zbl 1040.03017)], many-valued statements are considered and, moreover, the author gives an interpretation in terms of bets in the style of de Finetti. On the other hand, this approach does not allow to define conditional probabilities if the conditioning event is determined with probability zero.

The author shows that the whole investigation can be carried out in a logical and algebraic setting, and he finds a logical characterization of coherence for assessments of partially undetermined events.

The author shows that the whole investigation can be carried out in a logical and algebraic setting, and he finds a logical characterization of coherence for assessments of partially undetermined events.

Reviewer: Mirko Navara (Praha)

PDF
BibTeX
XML
Cite

\textit{F. Montagna}, J. Philos. Log. 41, No. 3, 563--593 (2012; Zbl 1261.03096)

Full Text:
DOI

##### References:

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

[2] | Blok, W., & Pigozzi, D. (1989). Algebraizable logics. In Memoirs of the American Mathematical Society (Vol. 396(77)). Providence: American Mathematical Society. · Zbl 0664.03042 |

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

[4] | Busaniche, M., & Cignoli, R. (2009). Residuated lattices as an algebraic semantics for paraconsistent Nelson’s logic. Journal of Logic and Computation, 19(6), 1019–1029. · Zbl 1191.03045 |

[5] | Canny, F. J. (1988). Some algebraic and geometric computation in PSPACE. In Proc. 20th ACM symp. on theory of computing (pp. 460–467). |

[6] | Chang, C. C. (1989). A new proof of the completeness of Łukasiewicz axioms. Transactions of the American Mathematical Society, 93, 74–80. · Zbl 0093.01104 |

[7] | Cintula, P. (2006). Weakly implicative (fuzzy) logics I: Basic properties. Archive for Mathematical Logic, 45, 673–704. · Zbl 1101.03015 |

[8] | Coletti, G., & Scozzafava, R. (2002). Probabilistic logic in a coherent setting. Dordrecht: Kluwer. · Zbl 1005.60007 |

[9] | Czelakowski, J. (2001). Protoalgebraic logics. Dordrecht: Kluwer. · Zbl 0984.03002 |

[10] | Dubois, D., & Prade, H. (1994). Conditional objects as nonmonotonic consequence relations. IEEE Transactions on Systems, Man and Cybernetics, 24(12), 1724–1740. IEEE. · Zbl 1371.03041 |

[11] | Flaminio, T., & Montagna, F. (2009). MV algebras with internal states and probabilistic fuzzy logics. International Journal of Approximate Reasoning, 50(1), 138–152. · Zbl 1185.06007 |

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

[13] | Kroupa, T. (2006). Every state on a semisimple MV algebra is integral. Fuzzy Sets and Systems, 157(20), 2771–2787. · Zbl 1107.06007 |

[14] | Kroupa, T. (2005). Conditional probability on MV-algebras. Fuzzy Sets and Systems, 149(2), 369–381. · Zbl 1061.60004 |

[15] | Kühr, J., & Mundici, D. (2007). De Finetti theorem and Borel states in [0,1]-valued algebraic logic. International Journal of Approximate Reasoning, 46(3), 605–616. · Zbl 1189.03076 |

[16] | Montagna, F. (2000). An algebraic approach to propositional fuzzy logic. Journal of Logic, Language and Information, 9, 91–124. · Zbl 0942.06006 |

[17] | Montagna, F. (2005). Subreducts of MV algebras with product and product residuation. Algebra Universalis, 53, 109–137. · Zbl 1086.06010 |

[18] | Montagna, F. (2009). A notion of coherence for books on conditional events in many-valued logic. Journal of Logic and Computation. doi: 10.1093/logcom/exp061 . First published online: 17 September 2009. · Zbl 1252.03042 |

[19] | Mundici, D. (1995). Averaging the truth value in Łukasiewicz logic. Studia Logica, 55(1), 113–127. · Zbl 0836.03016 |

[20] | Mundici, D. (2006). Bookmaking over infinite-valued events. International Journal of Approximate Reasoning, 46, 223–240. · Zbl 1123.03011 |

[21] | Mundici, D. (2008). Faithful and invariant conditional probability in Łukasiewicz logic. In D. Makinson, J. Malinowski, & H. Wansing (Eds.), Trends in logic 27: Towards mathematical philosophy (pp. 1–20). Springer. |

[22] | Panti, G. (2008). Invariant measures in free MV algebras. Communications in Algebra, 36, 2849–2861. · Zbl 1154.06008 |

[23] | Papadimitriu, C. H. (1994). Computational complexity. Addison-Wesley. |

[24] | Paris, J. B. (2001). A note on the Dutch Book Method. In T. S. Gert De Cooman, & T. Fine (Eds.), ISIPTA ’01, Proceedings of the second international symposium on imprecise probabilities and their applications, Ithaca, NY, USA. Shaker. |

[25] | Sendlewski, A. (1990). Nelson algebras through Heyting ones, I. Studia Logica, 49, 105–126. · Zbl 0714.06004 |

[26] | Tsinakis, C., & Wille, A. (2006). Minimal varieties of involutive residuated lattices. Studia Logica, 83, 407–423. · Zbl 1101.06010 |

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.