Asian Research Journal of Mathematics

Classically, formal individual choice and rationality are framed using precise preference relations. Individual preference relations and hence the rationality of choice are generally not precise, and this necessitates the studies under vague preference relations. Until now, available studies of rationality have done by utilizing R-implications in the associated continuous t-norm. However, another class of fuzzy implications, S-implications, which represent behavioral interpretation, i.e., lenience and non-satisfaction of criteria, have not yet been used for the study of rationality. We validated that S-implication with continuous t-norm fails to satisfy key results of rationality and hence introduced an operator called S-residuum, which logically represents the minimal degree required to hold the implication against the joint degree of satisfaction of the implication by t-norm. This happens due to failure of adjointness, the law of identity and the boundary condition in this structure. This S-implication-S-residuum framework is introduced and exploited to develop logical interrelations among rationality axioms built on vague preferences for the first time. This leads to reformulating the well-known rationality axioms, namely S-FWARP, S-FSARP, S-FWCA and S-FSCA. The main results are : S-FSCA ⇒ S-FWCA ⇒ S-FSWCA and S-FSARP ⇒ S-FWARP ⇒ S-SFWARP. Furthermore, we characterize regularity of fuzzy preference relation in terms of FCF satisfying S-FWCA and hence S-SFCA. We concluded that this new framework offers a behaviorally meaningful and logically coherent foundation for the study of vague choice and rationality.