Bilattices for deductions in multi-valued logic

In this exploratory paper we propose a framework for the deduction apparatus of multi-valued logics based on the idea that a deduction apparatus has to be a tool to manage information on truth values and not directly truth values of the formulas. This is obtained by embedding the algebraic structure V defined by the set of truth values into a bilattice B. The intended interpretation is that the elements of B are pieces of information on the elements of V. The resulting formalisms are particularized in the framework of fuzzy logic programming. Since we see fuzzy control as a chapter of multi-valued logic programming, this suggests a new and powerful approach to fuzzy control based on positive and negative conditions.

• We propose bilattice theory to distinguish the “truth dimension” from “information dimension” in multivalued logic.
• Elements of a bilattice are considered as “pieces of information” on truth values.
• We test our formalism considering two famous paradoxes: the heap paradox and the bald men paradox.
• Multivalued logic programming is the most promising field of application.
• The resulting approach to logic programming gives new powerful tools to fuzzy control.