Some abstract versions of G\"odel's second incompleteness theorem based   on non-classical logics
  We study abstract versions of G\"odel's second incompleteness theorem and formulate generalizations of L\"ob's derivability conditions that work for logics weaker than the classical one. We isolate the role of contraction rule in G\"odel's theorem and give a (toy) example of a system based on modal logic without contraction invalidating G\"odel's argument. 
Lev Beklemishev and Daniyar Shamkanov
math.LO