A Deterministic Weakening of Belnap–Dunn Logic

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Abstract

A deterministic weakening \(\mathsf {DW}\) of the Belnap–Dunn four-valued logic \(\mathsf {BD}\) is introduced to formalize the acceptance and rejection of a proposition at a state in a linearly ordered informational frame with persistent valuations. The logic \(\mathsf {DW}\) is formalized as a sequent calculus. The completeness and decidability of \(\mathsf {DW}\) with respect to relational semantics are shown in terms of normal forms. From an algebraic perspective, the class of all algebras for \(\mathsf {DW}\) is described, and found to be a subvariety of Berman’s variety \(\mathcal {K}_{1,2}\). Every linearly ordered frame is logically equivalent to its dual algebra. It is proved that \(\mathsf {DW}\) is the logic of a nine-element distributive lattice with a negation. Moreover, \(\mathsf {BD}\) is embedded into \(\mathsf {DW}\) by Glivenko’s double-negation translation.

Keywords

Belnap–Dunn logic Relational semantics Ockham algebras 

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Logic and CognitionSun Yat-sen UniversityGuangzhouChina

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