Notre Dame Journal of Formal Logic Volume 51, Number 4, 2010 A Note on Majki´ c’s Systems Hitoshi Omori and Toshiharu Waragai Abstract The present note offers a proof that systems developed by Majki´ c are actually extensions of intuitionistic logic, and therefore not paraconsistent. 1 Introduction In [3], Majki´ c developed two hierarchies of “paraconsistent” logic called Z n and CZ n (1 ≤ n <ω), which are variations of da Costa’s hierarchy C n (cf. da Costa [2]). As is mentioned in [3], this was motivated by the lack of “a kind of (relative) compositional model-theoretic semantics” (cf. [3, p. 404]) for da Costa’s systems. Now, the aim of the present note is to prove the following two facts. Fact 1.1 Two hierarchies Z n and CZ n are not actually a hierarchy in the sense that for any i  = j , Th( Z i ) = Th( Z j ) and Th(CZ i ) = Th(CZ j ) hold, where Th( S) stands for the set of theorems in a system S. Fact 1.2 Systems Z n and CZ n are not paraconsistent, but instead they are extended systems of intuitionistic propositional calculus. These will be proved by giving a simple axiomatization for Z n and CZ n which is different from the original one. 2 Formulation of Z n and CZ n We shall first revisit the systems Z n and CZ n . First, the positive part of these systems is intuitionistic; that is, it consists of the following axiom schemata and a rule of inference (we shall refer to this system as IPC + ): (1) A ⊃ ( B ⊃ A) (2) ( A ⊃ B ) ⊃ (( A ⊃ ( B ⊃ C )) ⊃ ( A ⊃ C )) (3) ( A ∧ B ) ⊃ A (4) ( A ∧ B ) ⊃ B Received October 13, 2009; accepted November 13, 2009; printed September 21, 2010 2010 Mathematics Subject Classification: Primary, 03B53 Keywords: paraconsistent logic, intuitionistic logic, Majki´ c’s systems Z n and CZ n c  2010 by University of Notre Dame 10.1215/00294527-2010-032 503