Remarks on the Symmetric Difference from an Inferential Point of View
Itziar Garcia-Honrado and Enric Trillas
This paper is devoted to the study of disjunctive syllogisms in different algebraic frameworks: Boolean algebras, De Morgan algebras, some cases of three-valued logic and algebras of fuzzy sets. It is considered the symmetric difference operator as a mathematical translation of the exclusive disjunction connective. Some properties of this operator are studied by comparing its behavior in the different mentioned frameworks. Furthermore, the main topic of this paper is in which cases the operators of symmetric difference allow to preserve the inferential schemes of disjunctive syllogism. In the case of fuzzy set algebras, it is obtained a boundary for any function able to translate the properties of the symmetric difference in order to keep the disjunctive syllogism. The paper tries to stress the inferential interest of the symmetric difference, and extends extends previous work on the subject.
Keywords: Exclusive disjunction; symmetric difference; inference; three valued logic; boolean algebra; De Morgan algebra and fuzzy logic