## Can We Enforce Full Compositionality in Uncertainty Calculi? (1994)

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Venue: | In Proc. of the 11th Nat. Conf. on Artificial Intelligence (AAAI-94 |

Citations: | 19 - 2 self |

### BibTeX

@INPROCEEDINGS{Dubois94canwe,

author = {Didier Dubois and Henri Prade},

title = {Can We Enforce Full Compositionality in Uncertainty Calculi?},

booktitle = {In Proc. of the 11th Nat. Conf. on Artificial Intelligence (AAAI-94},

year = {1994},

pages = {149--154},

publisher = {AAAI Press / MIT Press}

}

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### Abstract

At AAAI'93, Elkan has claimed to have a result trivializing fuzzy logic. This trivialization is based on too strong a view of equivalence in fuzzy logic and relates to a fully compositional treatment of uncertainty. Such a treatment is shown to be impossible in this paper. We emphasize the distinction between i) degrees of partial truth which are allowed to be truth functional and which pertain to gradual (or fuzzy) propositions, and ii) degrees of uncertainty which cannot be compositional with respect to all the connectives when attached to classical propositions. This distinction is exemplified by the difference between fuzzy logic and possibilistic logic. We also investigate an almost compositional uncertainty calculus, but it is shown to lack expressiveness. 1. Introduction There is a very active research trend in Artificial Intelligence concerning the management of uncertainty in knowledge-based systems. This trend is still influenced by the MYCIN experiments (Buchanan & Shortli...

### Citations

4028 | Fuzzy sets - Zadeh |

490 |
Fuzzy sets as the basis for a theory of possibility”, Fuzzy Sets and Systems
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- 1978
(Show Context)
Citation Context ...ed by means of fuzzy sets) on the other hand. The first problem can be addressed by means of a truth functional fuzzy (multiple-valued) logic, while the second one is the realm of possibility theory (=-=Zadeh, 1978-=-; Dubois & Prade, 1988a) which is a non-fully compositional uncertainty calculus (i.e., the degree of uncertainty of a compound proposition cannot systematically be computed from the degrees of uncert... |

441 |
Probabilistic logic
- Nilsson
- 1986
(Show Context)
Citation Context ...for computing the resulting weights in the style of many-valued logics where all the connectives are usually truth-functional. Even degrees of probability are sometimes called degrees of truth (e.g., =-=Nilsson, 1986-=-) although probabilistic logic excludes compositionality. The emergence of fuzzy rule-based systems in process control problems has led AI researchers, that criticized MYCIN-like systems, to reject fu... |

254 |
H.: Possibility Theory
- Dubois, Prade
- 1988
(Show Context)
Citation Context ...f fuzzy sets) on the other hand. The first problem can be addressed by means of a truth functional fuzzy (multiple-valued) logic, while the second one is the realm of possibility theory (Zadeh, 1978; =-=Dubois & Prade, 1988-=-a) which is a non-fully compositional uncertainty calculus (i.e., the degree of uncertainty of a compound proposition cannot systematically be computed from the degrees of uncertainty of its component... |

239 | H.: Possibilistic logic - Dubois, Prade - 1986 |

167 |
L.: A theory of approximate reasoning
- Zadeh
- 1979
(Show Context)
Citation Context ...ch truth-value a �� L, such that a = t d (S) a degree of possibility �� D (d) that a is the truth-value of S. This fuzzy set of more or less possible truth-values forms a so-called fuzzy truth=-=-value (Zadeh, 1979-=-) denoted t D (S). A fuzzy truth-value combines the ideas of partial truth and of uncertainty about truth. It is thus a more complex construct than degrees of truth and degrees of uncertainty. Changin... |

96 |
Theories of Probability
- Fine
- 1973
(Show Context)
Citation Context ...induce a comparative probability ordering on events. Namely a comparative probability ordering �� is such that �� is complete and transitive, S �� ��, " S �� W, and �� sat=-=isfies the additivity axiom (Fine, 1973): "S 1 , if S 1 �� (S 2 �-=-� S 3 ) = ��, then S 2 > S 3 �� S 1 �� S 2 > S 1 �� S 3 (14) where S 1 > S 2 means S 1 �� S 2 and not (S 2 �� S 1 ). Any non-degenerate function g classifies the events in W ... |

94 |
H.: An introduction to possibilistic and fuzzy logics. In: Non-standard logics for automated reasoning
- Dubois, Prade
- 1988
(Show Context)
Citation Context ...f fuzzy sets) on the other hand. The first problem can be addressed by means of a truth functional fuzzy (multiple-valued) logic, while the second one is the realm of possibility theory (Zadeh, 1978; =-=Dubois & Prade, 1988-=-a) which is a non-fully compositional uncertainty calculus (i.e., the degree of uncertainty of a compound proposition cannot systematically be computed from the degrees of uncertainty of its component... |

81 | The paradoxical success of fuzzy logic
- Elkan
- 1994
(Show Context)
Citation Context ...d except the laws of noncontradiction and of excluded middle (Bellman & Giertz, 1973). Sometimes, arguments against fuzzy set theory rely on the impossibility of compositionality (e.g., Weston, 1987; =-=Elkan, 1993-=-). Usually these arguments are based on the wrong assumption that the algebra of propositions to be evaluated is Boolean. Note that fuzzy truth values (case d above) are not truth- functional, general... |

74 |
Epistemic entrenchment and possibilistic logic
- Dubois, Prade
- 1991
(Show Context)
Citation Context ...mplete). This distinction is a crucial prerequisite in any discussion about fuzzy sets and possibility theory and their use in automated reasoning. Possibility measures have been shown (Dubois, 1986; =-=Dubois & Prade, 1991) to be the numerical co-=-unterpart of so-called qualitative possibility relations # (where S 1 # S 2 reads "S 1 is at least as possible as S 2 "), in the sense that "��, $ an ordering # such that "S 1 ... |

53 |
Belief structures, possibility theory and decomposable confidence measures on finite sets
- Dubois
- 1986
(Show Context)
Citation Context ...sumed to be complete). This distinction is a crucial prerequisite in any discussion about fuzzy sets and possibility theory and their use in automated reasoning. Possibility measures have been shown (=-=Dubois, 1986; Dubois & Prade, 1-=-991) to be the numerical counterpart of so-called qualitative possibility relations # (where S 1 # S 2 reads "S 1 is at least as possible as S 2 "), in the sense that "��, $ an orde... |

43 |
On the analytic formalism of the theory of fuzzy sets
- Bellman, Giertz
- 1973
(Show Context)
Citation Context ...ributive lattice structure that is compatible with the unit interval; this structure is the only one where all laws of Boolean algebra hold except the laws of noncontradiction and of excluded middle (=-=Bellman & Giertz, 1973-=-). Sometimes, arguments against fuzzy set theory rely on the impossibility of compositionality (e.g., Weston, 1987; Elkan, 1993). Usually these arguments are based on the wrong assumption that the alg... |

14 |
New results about properties and semantics of fuzzy set theoretic operators
- Dubois, Prade
- 1980
(Show Context)
Citation Context ...y of conjunction and disjunction. Indeed, the laws of excluded middle and noncontradiction are not consistent with idempotency of conjunction and disjunction, when truth is no longer a binary notion (=-=Dubois & Prade, 1980). Note that-=- Elkan (1993) finds it natural to require that propositions (A �� B) and B �� (A �� B) be equivalent, and shows that this requirement is incompatible with the convention of propositions ha... |

13 | Knowledge in Flux - Grdenfors - 1988 |

12 | A new normative theory of probabilistic logic - Aleliunas - 1988 |

11 | La logique de la probabilité - FINETTI - 1936 |

7 |
Approximate Truth
- Weston
- 1987
(Show Context)
Citation Context ... propositions S 1 , S 2 , S g(not S) = f(g(S)) (1) ; g(S 1 �� S 2 ) = g(S 1 ) * g(S 2 ) (2) g(S 1 �� S 2 ) = g(S 1 ) �� g(S 2 ) (3) This result is proved independently in (Dubois & Prade, =-=1988b) and (Weston, 1987-=-). A family of propositions represented by a classical language form a Boolean algebra. The lack of compositionality is then a direct consequence of the well-known fact in mathematics that a non-trivi... |

2 |
Evidence measures based on fuzzy information. Automatica
- Dubois, Prade
- 1985
(Show Context)
Citation Context ...t the state of facts is restricted by D. The fuzzy truth value �� t D (S) can be approximated by means of two numbers ��(S) and N(S), which extend (8) and its dual to the case of a fuzzy state=-=ment S (Dubois & Prade, 1985), namely with p = ���-=-� D ��(S) = sup d min(�� S (d), p(d)) N(S) = 1 -- ��(S) = inf d max(�� S (d), 1 -- p(d)). Indeed ��(S) and N(S) can be viewed as the degrees of possibility and necessity that S is ... |

1 |
Rule-Based Expert Systems. Readings
- Buchanan, Shortliffe
- 1984
(Show Context)
Citation Context ...oduction There is a very active research trend in Artificial Intelligence concerning the management of uncertainty in knowledge-based systems. This trend is still influenced by the MYCIN experiments (=-=Buchanan & Shortliffe, 1984-=-), where a basic idea was to attach weights expressing uncertainty to facts and rules in a knowledge base. Then we are faced with the problem of how to propagate these weights in reasoning procedures.... |

1 | A min-max semantics for fuzzy likelihood - Schwartz - 1992 |

1 | Knowledge in Flux - Gadenfors - 1988 |