YinYang bipolar logic and bipolar fuzzy logic

It is observed that equilibrium (including quasi- or non-equilibrium) is natural reality or bipolar truth. It is asserted that a multiple valued logic is a finite-valued extension of Boolean logic; a fuzzy logic is a real-valued extension of Boolean logic; Boolean logic and its extensions are unipolar systems that cannot be directly used to represent bipolar truth for visualization. To circumvent the representational limitations of unipolar systems, a zero-order (propositional) bipolar combinational logic BCL1 in the bipolar space B-1 = {-1, 0} x {0, 1} is upgraded to a first-order (predicate) bipolar logic. BCL1 is then extended to an (n + 1)(2) -valued crisp bipolar combinational logic BCLn in the bipolar space B-n = {-n,..., -2, - 1, 0} x {0, 1, 2,..., n} and a real-valued bipolar fuzzy logic BCLF in the bipolar space B-F = [- 1, 0] x [0, 1]. A bipolar counterpart of unipolar axioms and rules of inference is introduced with the addition of bipolar augmentation. First-order consistency and completeness are proved. Depolarization functions are identified for the recovery of BCL1, BCLn, and BCLF to Boolean logic, a (n + 1) -valued logic, and fuzzy logic, respectively. Thus, BCL1, BCLn, and BCLF are bipolar generalizations or fusions of Boolean logic, multiple valued logic, and fuzzy logic, respectively. The bipolar family of systems provides a unique representation for bipolar knowledge fusion and visualization in an equilibrium world. The semantics of the bipolar systems are established, justified, and compared with unipolar systems. A redress is presented for the ancient paradox of the liar that leads to a few comments on Godel's incompleteness theorem. (C) 2003 Elsevier Inc. All rights reserved.