A general method for deciding about logically constrained issues

Abstract

A general method is given for revising degrees of belief and arriving at consistent decisions about a system of logically constrained issues. In contrast to other works about belief revision, here the constraints are assumed to be fixed. The method has two variants, dual of each other, whose revised degrees of belief are respectively above and below the original ones. The upper [resp. lower] revised degrees of belief are uniquely characterized as the lowest [resp. greatest] ones that are invariant by a certain max-min [resp. min-max] operation determined by the logical constraints. In both variants, making balance between the revised degree of belief of a proposition and that of its negation leads to decisions that are ensured to be consistent with the logical constraints. These decisions are ensured to agree with the majority criterion as applied to the original degrees of belief whenever this gives a consistent result. They are also ensured to satisfy a property of respect for unanimity about any particular issue, as well as a property of monotonicity with respect to the original degrees of belief. The application of the method to certain special domains comes down to well established or increasingly accepted methods, such as the single-link method of cluster analysis and the method of paths in preferential voting.Mathematics Subject Classifications (2010) 03B42 · 68T37 · 91B06 · 91B14 · 91C20 R. Camps · X. Mora (B) · L. Saumell

Social choice rules driven by propositional logic

“…In this case, the basic decision associated with the upper revised valuation v * is the same. Theorem 2.6 (Respect for unanimity [8,Thm. 3.11]).…”

“…In particular, it can happen that it holds whenever p is accepted according to v * ; in that case we say that the doctrine is unquestionable for p when accepted. Sufficient conditions for ensuring such properties are given in [8,Thm. 4.8,Cor.…”

“…More specifically, we have restricted ourselves to a revision method that we introduced in [8] and that uses only the max and min operators. This automatically excludes such reputed rules as that of Borda and that of Condorcet, Kemény and Young [18,19,25].…”

“…In order to make sure that the upper revised valuations that we are dealing with are the right ones, we must use either the Blake canonical form or a * -equivalent one. According to [8,Cor. 4.6], in order to prove * -equivalence it suffices to check that the conjunctive normal form under consideration is disjoint-resolvable as defined in [8, § 4.2].…”

“…In this article we develop certain applications of a method for revising degrees of belief that we introduced in [8]. These applications belong to social choice theory.…”