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[[FILE:Venn1011.svg|border|right|thumb|240px|Implication]]
In 1993 Elkan identified a ''paradox'' that [https://en.wikipedia.org/wiki/Fuzzy_logic#Fuzzy_logic_operators fuzzy logic equivalentsoperators] (min, max, 1-x) of applied to the equivalence equation of two re-expressions of the [https://en.wikipedia.org/wiki/Material_implication_(rule_of_inference) material implication] a → b
~(a & ~b) = (~a & ~b) | b
that is
1 - max(a, 1-b) = max(min(1-a, 1-b), b)
is not true for if a < b < 0.5.
=Selected Publications=
==2000 ...==
* [[Charles Elkan]] ('''2001'''). ''The Foundations of Cost-Sensitive Learning''. [[Conferences#IJCAI|IJCAI 2001]]
* [[Charles Elkan]] ('''2001'''). ''Paradoxes of fuzzy logic, revisited''. [https://dblp.uni-trier.de/db/journals/ijar/ijar26.html International Journal of Approximate Reasoning, Vol. 26], No. 2], [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.20.7605 CiteSeerX]
* [[Mathematician#ETrillas|Enric Trillas]], [[Mathematician#CAlsina|Claudi Alsina]] ('''2001'''). ''Comments to "Paradoxes of fuzzy logic, revisited"''. [https://dblp.uni-trier.de/db/journals/ijar/ijar26.html International Journal of Approximate Reasoning, Vol. 26], No. 2, [https://core.ac.uk/download/pdf/82697342.pdf pdf]
* [[Mathematician#ETrillas|Enric Trillas]], [[Mathematician#CAlsina|Claudi Alsina]] ('''2001'''). ''Elkan's theoretical argument, reconsidered''. [https://dblp.uni-trier.de/db/journals/ijar/ijar26.html International Journal of Approximate Reasoning, Vol. 26], No. 2, [https://core.ac.uk/download/pdf/81951492.pdf pdf]
