Bipolar Sets and YinYang Bipolar Dynamic Logic (BDL)

In this chapter an equilibrium-based set-theoretic approach to mathematical abstraction and axiomatization is presented for resolving the LAFIP paradox (Ch. 1) and for enabling logically definable causality (Ch. 2). Bipolar set theory is formally presented, which leads to YinYang bipolar dynamic logic (BDL). BDL in zeroth-order, 1st-order, and modal forms are presented with four pairs of dynamic De Morgan’s laws and a bipolar universal modus ponens (BUMP). BUMP as a key element of BDL enables logically definable causality and quantum computing. Soundness and completeness of a bipolar axiomatization are asserted; computability is proved; computational complexity is analyzed. BDL can be considered a non-linear bipolar dynamic generalization of Boolean logic plus quantum entanglement. Despite its nonlinear bipolar dynamic quantum property, it does not compromise the basic law of excluded middle. The recovery of BDL to Boolean logic is axiomatically proved through depolarization and the computability of BDL is proved. A redress on the ancient paradox of the liar is presented with a few observations on Gödel’s incompleteness theorem. Based on BDL, bipolar relations, bipolar transitivity, and equilibrium relations are introduced. It is shown that a bipolar equilibrium relation can be a non-linear bipolar fusion of many equivalence relations. Thus, BDL provides a logical basis for YinYang bipolar relativity–an equilibrium-based axiomatization of social and physical sciences.