Upper Boundary Algebra for Modeling the Missing Values Is a Residuated Lattice

It is already more than 100 years since the first proposal on three-valued logic appeared and it became a seminal work initiating lots of followers among scholars and researchers. Since then, we have observed distinct logical and algebraic approaches to modeling undefined values, i.e., the situation when the truth value of a given proposition is neither true nor false, but it is not defined. These various algebraic models of three-valued functionality are built to model various types of undefinedness, e.g., conceptional undefinedness, inconsistencies, indeterminable values, meaningless values, or half-true. It is not surprising that recently, these three-valued logics have been extended to partial fuzzy logics, i.e. specific many-valued logics that are extended by the dummy value * that models the undefined truth value. The algebraic structures for such logics are called partial algebras. Recently, two partial algebras, namely the Dragonfly algebra and the Lower Estimation, were both developed to capture the missing or unknown values. Their main idea consists in determining the lower boundary of the truth value of a proposition that we may guarantee after processing the operations independently on what values would replace the dummies *. Such an approach naturally leads to the consequence that the dummy * behaves as a “nearly zero” or “almost false” value. Though the application potential of such algebras in processing the missing values turned out to be very useful at some problems, it turned to be promising to consider a nearly dual approach. Such an approach should model the upper boundary idea and lead to a “nearly one” or “almost true” value. This study provides the first definition of such an algebra and investigates which of the standard properties of residuated lattices remain preserved. Unlike in the lower boundary case, we surprisingly show that in principle all of them are preserved, i.e., that the Upper Boundary algebra, though extended, remains to be the residuated lattice.