As it seems that on the research level notation is the biggest problem for some of people, I will share a link to classical notation of Łukasiewicz L3 system, where deductive theorem i classical meaning does not hols ( but modified version - holds) - just take a look here: [L3 Łukasiewicz logic][1] and here: [Łukasiewicz L3 system][2]
and reference:  

> Bergmann, Merrie (2008). An introduction to many-valued and fuzzy
> logic: semantics, algebras, and derivation systems. Cambridge
> University Press. p. 114. ISBN 978-0-521-88128-9.


  [1]: https://en.wikipedia.org/wiki/Rule_of_inference#Example:_Hilbert_systems_for_two_propositional_logics
  [2]: https://en.wikipedia.org/wiki/Three-valued_logic#.C5.81ukasiewicz_logic