Lukasiewicz Negation and Many-Valued Extensions of Constructive Logics
Thomas Macaulay Ferguson · 2014
This paper examines the relationships between the many-valued logics G~and Gn~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 Gn~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 Gn~to complement the many-valued semantics.