Countably Many Weakenings of Belnap–Dunn Logic

  • Minghui Ma
  • Yuanlei LinEmail author


Every Berman’s variety \(\mathbb {K}_p^q\) which is the subvariety of Ockham algebras defined by the equation \({\sim ^{2p+q}}a = {\sim ^q}a\) (\(p\ge 1\) and \(q\ge 0\)) determines a finitary substitution invariant consequence relation \(\vdash _p^q\). A sequent system \(\mathsf {S}_p^q\) is introduced as an axiomatization of the consequence relation \(\vdash _p^q\). The system \(\mathsf {S}_p^q\) is characterized by a single finite frame \(\mathfrak {F}_p^q\) under the frame semantics given for the formal language. By the duality between frames and algebras, \(\mathsf {S}_p^q\) can be viewed as a \(4^{2p+q}\)-valued logic as it is characterized by a distributive lattice of \(4^{2p+q}\) elements with a unary operator. Moreover, a structural-rule-free, cut-free and terminating sequent system \(\mathsf {G}_p^q\) is established for \(\vdash _p^q\). The Craig interpolation property of \(\vdash _p^q\) is shown proof-theoretically utilizing \(\mathsf {G}_p^q\).


Belnap–Dunn logic Frame semantics Algebras Sequent system 


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This work was supported by the key project of National Social Science Found of China (Grant no. 18ZDA033). Thanks are given to the reviewers’ insightful and helpful comments on the revision of this paper. In particular, Remark 2.2, Remark 2.11 and some facts given in the conclusion are pointed out by the first reviewer.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Philosophy, Institute of Logic and CognitionSun Yat-sen UniversityGuangzhouChina

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