The Fmla-Fmla Axiomatizations of the Exactly True and Non-falsity Logics and Some of Their Cousins

  • Yaroslav Shramko
  • Dmitry Zaitsev
  • Alexander Belikov


In this paper we present a solution of the axiomatization problem for the Fmla-Fmla versions of the Pietz and Rivieccio exactly true logic and the non-falsity logic dual to it. To prove the completeness of the corresponding binary consequence systems we introduce a specific proof-theoretic formalism, which allows us to deal simultaneously with two consequence relations within one logical system. These relations are hierarchically organized, so that one of them is treated as the basic for the resulting logic, and the other is introduced as an extension of this basic relation. The proposed bi-consequences systems allow for a standard Henkin-style canonical model used in the completeness proof. The deductive equivalence of these bi-consequence systems to the corresponding binary consequence systems is proved. We also outline a family of the bi-consequence systems generated on the basis of the first-degree entailment logic up to the classic consequence.


First-degree entailment Exactly true logic Non-falsity logic Fmla-Fmla logical framework Bi-consequence system 


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We thank two anonymous reviewers for their valuable comments that greatly contributed to improving the final version of the paper.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of PhilosophyKryvyi Rih State Pedagogical UniversityKryvyi RihUkraine
  2. 2.Department of LogicLomonosov Moscow State UniversityMoscowRussia

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