@ARTICLE{FUZIMPL4.MIZ,
  AUTHOR = {Grabowski, Adam},
  TITLE = {On Fuzzy Negations and Laws of Contraposition.
     Lattice of Fuzzy Negations},
  DAY = {21},
  MONTH = {November},
  YEAR = {2023},
  ADDRESS1 = {Faculty of Computer Science\\University of Bia{\l}ystok\\Cio{\l}kowskiego 1M, 15-245 Bia{\l}ystok\\Poland},
  SECTION1 = {Laws of Contraposition},
  SECTION2 = {Fuzzy Negations Revisited},
  SECTION3 = {Proposition 1.5.3},
  SECTION4 = {Some Further Examples of Fuzzy Implications},
  SECTION5 = {Fuzzy Lattice Revisited},
  SUMMARY = {We continue in the Mizar system \cite{Mizar-State-2015} the formalization of
  fuzzy implications according to the book of Baczy\'nski and Jayaram
  ``Fuzzy Implications''.
  In this article we define the laws of contraposition with the connection to 
  various fuzzy negations.
  Additionally, we introduce the lattice of fuzzy negations and show its basic
  properties.
  This work is a continuation of the development
  of fuzzy sets in Mizar \cite{GrabowskiFuzzy:2013}.}}

