The concept of expressing truth values as real numbers in the range between 0 and 1 can bring to mind the possibility of using complex numbers to express truth values. These truth values would have an imaginary dimension, for example between 0 and i. Two- or higher-dimensional truth could potentially be useful in systems of paraconsistent logic. If practical applications were to arise for such systems, multidimensional infinite-valued logic could develop as a concept independent of real-valued logic.[11]
Lotfi A. Zadeh proposed a formal methodology of fuzzy logic and its applications in the early 1970s. By 1973, other researchers were applying the theory of Zadeh fuzzy controllers to various mechanical and industrial processes. The fuzzy modeling concept that evolved from this research was applied to neural networks in the 1980s and to machine learning in the 1990s. The formal methodology also led to generalizations of mathematical theories in the family of t-norm fuzzy logics.[12]
In infinitary logic, degrees of provability of propositions can be expressed in terms of infinite-valued logic that can be described via evaluated formulas, written as ordered pairs each consisting of a truth degree symbol and a formula.[17]
In mathematics, number-free semantics can express facts about classical mathematical notions and make them derivable by logical deductions in infinite-valued logic. T-norm fuzzy logics can be applied to eliminate references to real numbers from definitions and theorems, in order to simplify certain mathematical concepts and facilitate certain generalizations. A framework employed for number-free formalization of mathematical concepts is known as fuzzy class theory.[18]
Philosophical questions, including the Sorites paradox, have been considered based on an infinite-valued logic known as fuzzy epistemicism.[19] The Sorites paradox suggests that if adding a grain of sand to something that is not a heap cannot create a heap, then a heap of sand cannot be created. A stepwise approach toward a limit, in which truth is gradually "leaked", tends to refute that suggestion.[20]
In the study of logic itself, infinite-valued logic has served as an aid to understand the nature of the human understanding of logical concepts. Kurt Gödel attempted to comprehend the human ability for logical intuition in terms of finite-valued logic before concluding that the ability is based on infinite-valued logic.[21] Open questions remain regarding the handling, in natural language semantics, of indeterminate truth values.[22]
^"1.4.4 Defuzzification"(PDF). Fuzzy Logic. Swiss Federal Institute of Technology Zurich. 2014. p. 4. Archived from the original(PDF) on 2009-07-09. Retrieved 2018-05-16.
^Cignoli, R.; Esteva, F; Godo, L.; Torrens, A. (2000). "Basic Fuzzy Logic is the logic of continuous t-norms and their residua". Soft Computing. 4 (2): 106–112. doi:10.1007/s005000000044.
^"The moral: an adequate theory must allow our statements involving the notion of truth to be risky: they risk being paradoxical if the empirical facts are extremely (and unexpectedly) unfavorable. There can be no syntactic or semantic 'sieve' that will winnow out the 'bad' cases while preserving the 'good' ones. ... I am somewhat uncertain whether there is a definite factual question as to whether natural language handles truth-value gaps — at least those arising in connection with the semantic paradoxes — by the schemes of Frege, Kleene, van Fraassen, or perhaps some other." Kripke, Saul (1975). "Outline of a Theory of Truth"(PDF). The Journal of Philosophy. 72 (19): 690–716. doi:10.2307/2024634. JSTOR2024634.