Self-Referential Reasoning in the Light of Extended Truth Qualification Principle

Summary. The purpose of this paper is to formulate truth-value assignment to self-referential sentences via Zadeh’s truth qualification principle and to present new methods to assign truthvalues to them. Therefore, based on the truth qualification process, a new interpretation of possibilities and truth-values is suggested by means of type-2 fuzzy sets and then, the qualification process is modified such that it results in type-2 fuzzy sets. Finally, an idea of a comprehensive theory of type-2 fuzzy possibility is proposed. This approach may be unified with Zadeh’s Generalized Theory of Uncertainty (GTU) in the future. In the era of information and communications, truth and reliability play a significant role and intelligent machinery are demanding new tools of reliable data interpretation and processing. Now, human beings tend to develop new means to handle the great bulk of available information. Unquestionably, mimicry of human’s ability to process the incoming data intelligently and deduce conclusions about its reliability as well as “meaning” is of equal importance with obtaining information, if not more important. After the development of fuzzy computation theory, the great contribution of L. A. Zadeh [1] revealed the fact that there might be rigorous tools via which the “meaning” of the analyzed information could be manipulated besides its statistical nature. He investigated the “meaningoriented” approach to information processing via the well-founded theory of fuzzy computation and established a new theory of possibility as a counterpart of the probability theory. One of the novel applications of possibility theory is the resolution of the liar paradox (and henceforth, self-referential sentences bearing a paradox). Zadeh proposed a method to assign a truth-value to the liar sentence [2, 3]. In this paper, we investigate the application of Zadeh’s method to resolve the paradox borne by self-referential sentences and propose an extension to Zadeh’s method, such that it can be useful in a broader sense. This extension is based on the concept of type-2 fuzzy sets. We first introduce type-2 fuzzy sets, possibility theory, and the concept of self-reference, and then based on Zadeh’s method we try to assign truth-values to self-referential sentences. In the next step, we extend Zadeh’s truth qualification principle to handle the existing uncertainties in fuzzy possibilities. Finally, a comprehensive theory of type-2 fuzzy possibility is touched.