Abstract
An implication in fuzzy logic, commonly defined as a two-place operation on the unit interval, is an extension of the classical binary implication. It plays important roles in both mathematical and applied sides of fuzzy set theory. Besides the basic properties, there are many potential properties for implications, among which eight are widely used in the literature. Different implications satisfying different subgroups of these eight properties can be found. However, certain interrelationships exist between these eight properties. This chapter aims to lay bare the interrelationships between these eight properties.When searching counterexamples to prove the independencies we discover a new class of implications determined only by a negation. We then examine under which conditions the eight properties are satisfied. Finally, we obtain the intersection of the new class of implications with the S- and R- implications.
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Shi, Y., Van Gasse, B., Ruan, D. (2010). Implications in Fuzzy Logic: Properties and a New Class. In: Cornelis, C., Deschrijver, G., Nachtegael, M., Schockaert, S., Shi, Y. (eds) 35 Years of Fuzzy Set Theory. Studies in Fuzziness and Soft Computing, vol 261. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16629-7_5
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DOI: https://doi.org/10.1007/978-3-642-16629-7_5
Publisher Name: Springer, Berlin, Heidelberg
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