Lukasiewicz Negation and Many-Valued Extensions of Constructive Logics

This paper examines the relationships between the many-valued logics G^~ and G_n^~ of Esteva, Godo, Hajek, and Navara, i.e., Godel logic G enriched with Łukasiewicz negation, and neighbors of intuitionistic logic. The popular fragments of Rauszer´s Heyting-Brouwer logic HB admit many-valued extensions similar to G which may likewise be enriched with Łukasiewicz negation; the fuzzy extensions of these logics, including HB, are equivalent to G ^~, as are their n-valued extensions equivalent to G_n^~ for any n ≥ 2. These enriched systems extend Wansing´s logic I₄C₄, showing that Łukasiewicz negation is a species of Nelson´s negation of constructible falsity and yielding a Kripke-style semantics for G^~ and G_n^~ to complement the many-valued semantics.