Fuzzy Systems and its Applications

Fuzzy Systems and its Applications

Fuzzy Logic on Nonstandard Unit Interval

Document Type : Original Article

Authors
Morteza Moniri 1
Mohammad Ammar Amini 2
1 Mathematics, Mathematical Sciences,Tehram, Iran
2 Shahid Beheshti University
10.22034/jfsa.2025.545496.1283
Abstract
In this paper, we extend Łukasiewicz logic from the standard interval $[0, 1]$ to the nonstandard interval ${}^*[0, 1]$. This interval includes not only real numbers between zero and one, but also numbers infinitely close to them. First, we will review fuzzy logic from a mathematical perspective and also the fundamentals of nonstandard analysis. Then, using the transfer principle in nonstandard analysis, we show that the algebraic and logical properties of MV are preserved in the nonstandard interval. Therefore, without the need to change the axioms or rules of inference, nonstandard values can be used to model linguistic concepts such as "negligible" or "largely acceptable". In addition, the application potential of this framework in areas such as expert systems, fuzzy control, and artificial intelligence is significant.
Keywords
fuzzy logic
nonstandard analysis
hyperreal numbers
soundness and completeness
Subjects
 
[3]    Bergmann, M. (2008). An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Derivation Systems, Cambridge University Press.
[4]    Castaño, D. & Varela, J. P. D. & Savoy, G. (2025), ”Strong completeness for the predicate logic of the continuous t-norms”, Fuzzy Sets and Systems, 500, pp. 1-25.
[5]    Goldblatt, R. (1998). Lectures on the hyperreals: an introduction to nonstandard analysis (Vol. 188). Springer Science & Business Media.
[6]    Hájek, P. (2001). Metamathematics of fuzzy logic (Vol. 4). Springer Science & Business Media.
[7]    Kanovei, V. & Reeken, M. (2004), Nonstandard Analysis, Axiomatically, Book Series Springer Monographs in Mathematics, Springer Berlin: Heidelberg.
[8]    Nelson, E. (1977). ”Internal Set Theory: A New Approach to Nonstandard Analysis,” Bulletin of the American Mathematical Society, 83 (6).
[9]    Nguyen, H. T., Walker, C., & Walker, E. A. (2018). A First Course in Fuzzy Logic, CRC Press.
[10]    Loeb, P. A. & Wolff, M. P. H. (eds.) (2015). Nonstandard Analysis for the Working Mathematician, Springer Dordrecht.
[11]    Robinson, A. (1974). Non-standard analysis. Princeton University Press.
[12]    Verbruggen, H. B. & Bruijn, P. M. (1997). ”Fuzzy control and conventional control: What is (and can be) the real contribution of Fuzzy Systems?”, Fuzzy Sets and Systems, 90 (2), pp. 151–160.
 
 
 
Fuzzy Systems and its Applications
Volume 8, Issue 2 - Serial Number 17
Open Access Statement
December 2025
Pages 109-125

  • Receive Date 06 September 2025
  • Revise Date 08 October 2025
  • Accept Date 19 October 2025