YinYang bipolar logic and bipolar fuzzy logic

Abstract

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 BCL 1 in the bipolar space B 1={−1,0}×{0,1} is upgraded to a first-order (predicate) bipolar logic. BCL 1 is then extended to an ( n+1) 2-valued crisp bipolar combinational logic BCL n in the bipolar space B n ={− n,…,−2,−1,0}×{0,1,2,…, n} and a real-valued bipolar fuzzy logic BCL F in the bipolar space B F=[−1,0]×[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 BCL 1, BCL n , and BCL F to Boolean logic, a ( n+1)-valued logic, and fuzzy logic, respectively. Thus, BCL 1, BCL n , and BCL F 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 Gödel's incompleteness theorem.