Saturday, May 22, 2010

#92. Old paper by Dubois and Prade

Since everybody knows Dubois and Prade, and knows their work is available at the IRIT website, my linking to their papers is not really necessary.

The main points of the following paper are (assumed to be) well-known for insiders. Unfortunately, the confusions the authors patiently tried to dispel remain alive and kicking today.

Didier Dubois, Henri Prade (2001). Possibility theory, probability theory and multiple-valued logics: a clarification. Annals of Mathematics and Artificial Intelligence 32, 35-66.

The abstract is self-explanatory: `There has been a long-lasting misunderstanding in the literature of artificial intelligence and uncertainty modeling, regarding the role of fuzzy set theory and many-valued logics. The recurring question is that of the mathematical and pragmatic meaningfulness of a compositional calculus and the validity of the excluded middle law. This confusion pervades the early developments of probabilistic logic, despite early warnings of some philosophers of probability. This paper tries to clarify this situation. It emphasizes three main points. First, it suggests that the root of the controversies lies in the unfortunate confusion between degrees of belief and what logicians call "degrees of truth". The latter are usually compositional, while the former cannot be so. This claim is first illustrated by laying bare the non-compositional belief representation embedded in the standard propositional calculus. It turns out to be an all-or-nothing version of possibility theory. This framework is then extended to discuss the case of fuzzy logic versus graded possibility theory. Next, it is demonstrated that any belief representation where compositionality is taken for granted is bound to at worst collapse to a Boolean truth assignment and at best to a poorly expressive tool. Lastly, some claims pertaining to an alleged compositionality of possibility theory are refuted, thus clarifying a pervasive confusion between possibility theory axioms and fuzzy set basic connectives.'


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