Two Fuzzy Logic Programming Paradoxes (original) (raw)
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Two Fuzzy Logic Programming Paradoxes Imply Continuum Hypothesis
2008
Two different paradoxes of the fuzzy logic programming system of [29] are presented. The first paradox is due to two distinct (contradictory) truth values for every ground atom of FLP, one is syntactical, the other is semantical. The second paradox concerns the cardinality of the valid FLP formulas which is found to have contradictory values: both aleph_0\aleph_0aleph_0 the cardinality of the natural numbers, and ccc, the cardinality of the continuum. The result is that CH="False" and Axiom of Choice="False". Hence, ZFC is inconsistent.
2008
Two different paradoxes of the fuzzy logic programming system of [29] are presented. The first paradox is due to two distinct (contradictory) truth values for every ground atom of FLP, one is syntactical, the other is semantical. The second paradox concerns the cardinality of the valid FLP formulas which is found to have contradictory values: both aleph_0\aleph_0aleph_0 the cardinality of the natural numbers, and ccc, the cardinality of the continuum. The result is that CH="False" and Axiom of Choice="False". Hence, ZFC is inconsistent.
The paradoxical success of fuzzy logic
IEEE Expert, 2000
Fuzzy logic methods have been used successfully in many real-world applications, but the foundations of fuzzy logic remain under attack. Taken together, these two facts constitute a paradox. A second paradox is that almost all of the successful fuzzy logic applications are embedded controllers, while most of the theoretical papers on fuzzy methods deal with knowledge representation and reasoning. I hope here to resolve these paradoxes by identifying which aspects of fuzzy logic render it useful in practice, and which aspects are inessential. My conclusions are based on a mathematical result, on a survey of literature on the use of fuzzy logic in heuristic control and in expert systems, and on practical experience developing expert systems.
submitted to the Fuzzy Sets and Systems Journal
2012
Two different paradoxes of the fuzzy logic programming system of [29] are presented. The first paradox is due to two distinct (contradictory) truth values for every ground atom of FLP, one is syntactical, the other is semantical. The second paradox concerns the cardinality of the valid FLP formulas which is found to have contradictory values: both ℵ0 the cardinality of the natural numbers, and c, the cardinality of the continuum. 1.
The sorites paradox and fuzzy logic
International Journal of General Systems, 2003
The sorites paradox (interpreted as the paradox of small natural numbers) is analyzed using mathematical fuzzy logic. In the first part, we present an extension of BL-fuzzy logic by a new unary connective At of almost true and the crisp Peano arithmetic extended by a fuzzy predicate of feasibility. Then we give examples of possible semantics of At and examples of semantics of feasible numbers. In the second part, we present an analysis of the sorites paradox within fuzzy logic with evaluated syntax and show that under a very natural assumption we obtain a consistent fuzzy theory. Thus, sorites is not paradoxical at all.
The Kleene-Rosser Paradox, The Liar's Paradox & A Fuzzy Logic Programming Paradox Imply SAT is (NO
2008
After examining the {\bf P} versus {\bf NP} problem against the Kleene-Rosser paradox of the lambda\lambdalambda-calculus [94], it was found that it represents a counter-example to NP-completeness. We prove that it contradicts the proof of Cook's theorem. A logical formalization of the liar's paradox leads to the same result. This formalization of the liar's paradox into a computable form is a 2-valued instance of a fuzzy logic programming paradox discovered in the system of [90]. Three proofs that show that {\bf SAT} is (NOT) NP-complete are presented. The counter-example classes to NP-completeness are also counter-examples to Fagin's theorem [36] and the Immermann-Vardi theorem [89,110], the fundamental results of descriptive complexity. All these results show that {\bf ZF$\not$C} is inconsistent.
Quantitative Study of Fuzzy Logics
2020 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 2020
In this paper, we focus on two main 3-valued logics used by the fuzzy logic community. The Gödel-Dummett logic and the Łukasiewicz one. Both are based on the same language of implication and negation. In both, we consider fragments consisting of formulas formed with one variable. We investigate the proportion of the number of true (or satisfiable) formulas of a certain length n to the number of all formulas of such length. We are especially interested in the asymptotic behavior of this fraction when length n tends to infinity. If the limit exists it is represented by a real number between 0 and 1 which is called the density of truth or the density of SAT. Using the powerful theory of analytic combinatorics, we state several results comparing the density of truth and the density of satisfiable formulas for both Gödel-Dummett and Łukasiewicz logics.
On Some Alleged Misconceptions about Fuzzy Logic
Artificial Intelligence Review - AIR, 2004
Entemann (2002) defends fuzzy logic by pointing to what he calls “misconceptions” concerning fuzzy logic. However, some of these ‘;misconceptions’ are in fact truths, and it is Entemann who has the misconceptions. The present article points to mistakes made by Entemann in three different areas. It closes with a discussion of what sort of general considerations it would take to motivate fuzzy logic.