Possibility based modal semantics for graded modifiers

Abstract

A brief introduction to basic modifiers is given. Any modifier with its dual and the corresponding negation form a DeMorgan triple similar to that of t-norms, t-conorms, and negation. The lattice structure of the unit interval with the usual partial order is similar to that of the set of all membership functions. This structure has a certain connection to implication, by means of the subsethood of fuzzy sets, and it is possible to create a similar expression for modifiers as the axiom of reflexivity is in modal logic. Also, other connections to modal logics can be found. This motivates to develop a formal semantics to modifier logic by means of that of modal logic. Actually, this kind of logic is so-called metalogic concerning either true or false statements about properties of modifiers. This version is based on graded possibility operations. Hence, semantic tools for weakening modifiers are derived. The corresponding things for substantiating modifiers are constructed by means of duality. Finally, some outlines for modifier systems are considered. © Springer-Verlag Berlin Heidelberg 2007.