RU and (U, N)-implications satisfying Modus Ponens

In this paper it is investigated when some kinds of fuzzy implication functions derived from uninorms satisfy the Modus Ponens with respect to a continuous t-norm T, or equivalently, when they are T-conditionals. The study is done for RU-implications and ( U , N ) -implications with N a continuous fuzzy negation leading to a lot of solutions in both cases. For RU-implications T-conditionality only depends on the underlying t-norm of the uninorm used to derive the residual implication. On the contrary, for ( U , N ) -implications the underlying t-norm is never relevant and only the region out of the t-norm is so. Even the t-conorm can be not relevant also in some cases. Characterization of all residual implications for uninorms satisfying Modus Ponens.Study of Modus Ponens for ( U , N ) -implications.Modus Ponens (T-conditionality) for implications generated from uninorms.

[1]  Michal Baczynski,et al.  Fuzzy Implications , 2008, Studies in Fuzziness and Soft Computing.

[2]  M. Mas,et al.  Migrative uninorms and nullnorms over t-norms and t-conorms , 2015, Fuzzy Sets Syst..

[3]  Joan Torrens,et al.  A characterization of (U, N), RU, QL and D-implications derived from uninorms satisfying the law of importation , 2010, Fuzzy Sets Syst..

[4]  Enric Trillas,et al.  When (S, N)-implications are (T, T1)-conditional functions? , 2003, Fuzzy Sets Syst..

[5]  Joan Torrens,et al.  A Survey on Fuzzy Implication Functions , 2007, IEEE Transactions on Fuzzy Systems.

[6]  Bernard De Baets,et al.  Residual operators of uninorms , 1999, Soft Comput..

[7]  Joan Torrens,et al.  Residual implications derived from uninorms satisfying Modus Ponens , 2015, IFSA-EUSFLAT.

[8]  Joan Torrens,et al.  A characterization of residual implications derived from left-continuous uninorms , 2010, Inf. Sci..

[9]  Bernard De Baets,et al.  Some Remarks on the Characterization of Idempotent Uninorms , 2010, IPMU.

[10]  Joan Torrens,et al.  S- and R-implications from uninorms continuous in ]0, 1[2 and their distributivity over uninorms , 2009, Fuzzy Sets Syst..

[11]  Huawen Liu,et al.  Single-Point Characterization of Uninorms with Nilpotent Underlying T-Norm and T-Conorm , 2014, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[12]  Eloy Renedo,et al.  Extracting causation knowledge from natural language texts , 2005 .

[13]  Humberto Bustince Sola,et al.  Advances in Fuzzy Implication Functions , 2013 .

[14]  Joan Torrens,et al.  Modus ponens and modus tollens in discrete implications , 2008, Int. J. Approx. Reason..

[15]  Daniel Ruiz,et al.  Distributivity and conditional distributivity of a uninorm and a continuous t-conorm , 2006, IEEE Transactions on Fuzzy Systems.

[16]  Joan Torrens,et al.  On locally internal monotonic operations , 2003, Fuzzy Sets Syst..

[17]  Joan Torrens,et al.  Residual implications and co-implications from idempotent uninorms , 2004, Kybernetika.

[18]  Joan Torrens,et al.  Two types of implications derived from uninorms , 2007, Fuzzy Sets Syst..

[19]  Francesc Esteva,et al.  Review of Triangular norms by E. P. Klement, R. Mesiar and E. Pap. Kluwer Academic Publishers , 2003 .

[20]  Bernard De Baets,et al.  A single-point characterization of representable uninorms , 2012, Fuzzy Sets Syst..

[21]  E. Trillas,et al.  On MPT-implication functions for Fuzzy Logic , 2004 .

[22]  Ronald R. Yager,et al.  Structure of Uninorms , 1997, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[23]  Manuel González Hidalgo,et al.  On the Choice of the Pair Conjunction–Implication Into the Fuzzy Morphological Edge Detector , 2015, IEEE Transactions on Fuzzy Systems.

[24]  Shi-kai Hu,et al.  The structure of continuous uni-norms , 2001, Fuzzy Sets Syst..

[25]  Huawen Liu,et al.  Distributivity and conditional distributivity of a uninorm with continuous underlying operators over a continuous t-conorm , 2016, Fuzzy Sets Syst..

[26]  Joan Torrens,et al.  On the representation of fuzzy rules , 2008, Int. J. Approx. Reason..

[27]  E. Trillas,et al.  When QM‐operators are implication functions and conditional fuzzy relations , 2000 .

[28]  Balasubramaniam Jayaram,et al.  (U, N)-Implications and Their Characterizations , 2009, EUSFLAT Conf..

[29]  Joan Torrens,et al.  A survey on the existing classes of uninorms , 2015, J. Intell. Fuzzy Syst..

[30]  Joan Torrens,et al.  An Overview of Construction Methods of Fuzzy Implications , 2013 .