Kabbalah Logic and Semantic Foundations for a Postmodern Fuzzy Set and Fuzzy Logic Theory

Abstract

Despite half a century of fuzzy sets and fuzzy logic progress, as fuzzy sets address complex and uncertain information through the lens of human knowledge and subjectivity, more progress is needed in the semantics of fuzzy sets and in exploring the multi-modal aspect of fuzzy logic due to the different cognitive, emotional and behavioral angles of assessing truth. We lay here the foundations of a postmodern fuzzy set and fuzzy logic theory addressing these issues by deconstructing fuzzy truth values and fuzzy set membership functions to re-capture the human knowledge and subjectivity structure in membership function evaluations. We formulate a fractal multi-modal logic of Kabbalah which integrates the cognitive, emotional and behavioral levels of humanistic systems into epistemic and modal, deontic and doxastic and dynamic multi-modal logic. This is done by creating a fractal multi-modal Kabbalah possible worlds semantic frame of Kripke model type. The Kabbalah possible worlds semantic frame integrates together both the multi-modal logic aspects and their Kripke possible worlds model. We will not focus here on modal operators and axiom sets. We constructively define a fractal multi-modal Kabbalistic L-fuzzy set as the central concept of the postmodern fuzzy set theory based on Kabbalah logic and semantics.

Postmodern Fuzzy System Theory: A Deconstruction Approach Based on Kabbalah

“…Each sefirot unit is made up similarly of nine sub-sefirot connected according to a Tree of Life structure. (Source: [9]).…”

“…We introduce next the postmodern fuzzy set version, Kabbalistic fuzzy sets, the next step in our construction process of a postmodern fuzzy system theory described in Figure 1. Our approach to create a Kabbalah fuzzy set theory [9] is based on the "possible worlds" semantic model of modal logic [14], which is used to construct fuzzy set membership functions [15,16]. We will develop a modified Kabbalah fuzzy set theory in this paper that is better suitable for fuzzy system theory: a new hierarchic Kabbalah possible worlds model will be developed here based on Figure 3 with a new possible world "importance" weighting function based on the structure of the graph in Figure 3 [0,1] will be used instead of a lattice L to make calculations more intuitive.…”

“…The Kabbalah Tree of Life model offers a fractal structural model for each of these sub-systems. In order to open the door of KST to the complexity and imprecision of humanistic system applications we developed Kabbalistic fuzzy sets (KFS) [9] as a deconstructed version of fuzzy set theory [3], a postmodern fuzzy set theory. Fuzzy set theory is based on a fuzzy membership function taking nuanced truth-values due to human subjectivity, yet we see nowhere human cognitive, emotional and behavioral aspects featuring in the fuzzy membership function formulas.…”