The Fuzzy Completeness Theory

Abstract

The Two Incompleteness Theorems of Kurt Friedrich Gödel and the Impossibility Theorem of Kenneth Arrow claim that logic, the most reliable of human knowledge, is incomplete or can be inconsistent. The Fuzzy Completeness Theory states that the Fuzzy Logic of Lotfi A. Zadeh has resolved the incompleteness and impossibility in logic and made logic complete and knowledge reliable with the new concept of Range of Tolerance, within which logic is still complete and knowledge, valid. In the Age of Reason about 300 years ago just prior to the Age of Science, reasoning is free for all, without the constraint of the laws of nature, which would be discovered in the Age of Science. However, the Scientific Method of reasoning by empirical verification depends so much on faith that it is logically and empirically dismissed by mathematicians and logicians, especially, after the exposure by Thomas Kuhn and Paul Feyerabend that a scientific advancement is akin to a religious conversion. On the other hand, mathematicians and logicians have been working steadily to find the limit of reliable knowledge. In the current state of knowledge, Kurt Gödel has the last word with his Two Incompleteness Theorems, which conclude that the most reliable of human knowledge, logic, is incomplete, casting doubt whether knowledge is completely reliable. Gödel’s view is further supported by the Impossibility Theorem of Kenneth Arrow. However, Zadeh and the author of this paper extend Zadeh’s concept of Range of Value in Fuzzy Logic to that of Range of Tolerance. Accordingly, Fuzzy Logic deals with the sacrifice of precision in the process of expanding the Range of Tolerance of a creation in order for the creation to survive and flourish for all the possibility of an uncertain future.

In knowledge, incompleteness in logic can be resolved by the Range of Tolerance covering the incomplete part or ignoring the infrequent impossibilities, and, thus, making logic valid, again. Knowledge is derived generally from reason. Technically, the Fuzzy Completeness Theory classifies 16 Methods of Reason. The 16 Methods are the combination of the 4 basic Methods of Reason: 1) Logic, 2) Mathematics, 3) Empirical Verification, and 4) Others, each of which has 2 forms: 1) Fuzzy and 2) Exact and two types: 1) Complete and 2) Incomplete. Gödel, Arrow, and the Author agree that no matter how rigorous is the Method of Reason the reason cannot be complete, when the reason is Exact. When a solution is newly defined as an answer within the Range of Tolerance of the solution, Fuzzy Logic resolves the incompleteness in logic and becomes the new foundation of knowledge, replacing Exact Logic. With this definition of a solution, Fuzzy Logic covers the incomplete or the impossible parts of the solution by expanding sufficiently the Range of Tolerance to make reason complete and knowledge reliable, but only within the Range of Tolerance.

To summarize, even though the world’s leading intellectuals have proven, directly, that logic is incomplete and, indirectly, that knowledge is invalid, reality is still operating smoothly, and science has even demonstrated the power of knowledge. The conflict between the most reliable knowledge, namely, logic and the real world is resolved by Fuzzy Logic, which introduces the new concept of Range of Tolerance, within which reality can still operate in accordance with the laws discovered by knowledge. In sum, reality is fuzzy, not exact.

The breakthrough impact of this paper centers around completeness theory and Fuzzy Logic. In the early 21st century, the mainstream knowledge is still not aware that the supply and demand model is incomplete, and that the DNA-protein system resembles computer science based on logic more than science based on experimentation. The current computer is based on exact logic and is designed for temporary existence, while the living system is design for permanent existence and must depend on the Range of Tolerance based on Fuzzy Logic to survive permanently in an uncertain future. Financial crises will be caused by the unstable investment return, which is the incomplete part in the supply demand model. Complexity crises will be caused by the lack of the requirement of permanence or complete automation, which is the ultimate solution to unlimited complexity. The 16 Methods of Reason correspond roughly to Culture Level Quotient (CLQ), which is a non-technical measure of a person, a people or a nation.